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# Permutations

The Permutations module. Use `Permutation?` to get information about the Permutation class, and `Permutations?` to get information about the combinatorial class of permutations.

Warning

This file defined `Permutation` which depends upon `CombinatorialObject` despite it being deprecated (see trac ticket #13742). This is dangerous. In particular, the `Permutation._left_to_right_multiply_on_right()` method (which can be called trough multiplication) disables the input checks (see `Permutation()`). This should not happen. Do not trust the results.

## What does this file define ?

The main part of this file consists in the definition of permutation objects, i.e. the `Permutation()` method and the `Permutation` class. Global options for elements of the permutation class can be set through the `PermutationOptions` object.

Below are listed all methods and classes defined in this file.

Methods of Permutations objects

`left_action_product()` | Returns the product of `self` with another permutation, in which the other permutation is applied first.
`right_action_product()` | Returns the product of `self` with another permutation, in which `self` is applied first.
`size()` | Returns the size of the permutation `self`.
`cycle_string()` | Returns the disjoint-cycles representation of `self` as string.
`next()` | Returns the permutation that follows `self` in lexicographic order (in the same symmetric group as `self`).
`prev()` | Returns the permutation that comes directly before `self` in lexicographic order (in the same symmetric group as `self`).
`to_tableau_by_shape()` | Returns a tableau of shape `shape` with the entries in `self`.
`to_cycles()` | Returns the permutation `self` as a list of disjoint cycles.
`to_permutation_group_element()` | Returns a `PermutationGroupElement` equal to `self`.
`signature()` | Returns the signature of the permutation `sef`.
`is_even()` | Returns `True` if the permutation `self` is even, and `False` otherwise.
`to_matrix()` | Returns a matrix representing the permutation `self`.
`rank()` | Returns the rank of `self` in lexicographic ordering (on the symmetric group containing `self`).
`to_inversion_vector()` | Returns the inversion vector of a permutation `self`.
`inversions()` | Returns a list of the inversions of permutation `self`.
`show()` | Displays the permutation as a drawing.
`number_of_inversions()` | Returns the number of inversions in the permutation `self`.
`noninversions()` | Returns the `k`-noninversions in the permutation `self`.
`number_of_noninversions()` | Returns the number of `k`-noninversions in the permutation `self`.
`length()` | Returns the Coxeter length of a permutation `self`.
`inverse()` | Returns the inverse of a permutation `self`.
`ishift()` | Returns the `i`-shift of `self`.
`iswitch()` | Returns the `i`-switch of `self`.
`runs()` | Returns a list of the runs in the permutation `self`.
`longest_increasing_subsequence_length()` | Returns the length of the longest increasing subsequences of `self`.
`longest_increasing_subsequences()` | Returns the list of the longest increasing subsequences of `self`.
`cycle_type()` | Returns the cycle type of `self` as a partition of `len(self)`.
`foata_bijection()` | Returns the image of the permutation `self` under the Foata bijection \(\phi\).
`to_lehmer_code()` | Returns the Lehmer code of the permutation `self`.
`to_lehmer_cocode()` | Returns the Lehmer cocode of `self`.
`reduced_word()` | Returns the reduced word of the permutation `self`.
`reduced_words()` | Returns a list of the reduced words of the permutation `self`.
`reduced_word_lexmin()` | Returns a lexicographically minimal reduced word of a permutation `self`.
`fixed_points()` | Returns a list of the fixed points of the permutation `self`.
`number_of_fixed_points()` | Returns the number of fixed points of the permutation `self`.
`recoils()` | Returns the list of the positions of the recoils of the permutation `self`.
`number_of_recoils()` | Returns the number of recoils of the permutation `self`.
`recoils_composition()` | Returns the composition corresponding to the recoils of `self`.
`descents()` | Returns the list of the descents of the permutation `self`.
`idescents()` | Returns a list of the idescents of `self`.
`idescents_signature()` | Returns the list obtained by mapping each position in `self` to \(-1\) if it is an idescent and \(1\) if it is not an idescent.
`number_of_descents()` | Returns the number of descents of the permutation `self`.
`number_of_idescents()` | Returns the number of idescents of the permutation `self`.
`descents_composition()` | Returns the composition corresponding to the descents of `self`.
`descent_polynomial()` | Returns the descent polynomial of the permutation `self`.
`major_index()` | Returns the major index of the permutation `self`.
`imajor_index()` | Returns the inverse major index of the permutation `self`.
`to_major_code()` | Returns the major code of the permutation `self`.
`peaks()` | Returns a list of the peaks of the permutation `self`.
`number_of_peaks()` | Returns the number of peaks of the permutation `self`.
`saliances()` | Returns a list of the saliances of the permutation `self`.
`number_of_saliances()` | Returns the number of saliances of the permutation `self`.
`bruhat_lequal()` | Returns `True` if self is less or equal to `p2` in the Bruhat order.
`weak_excedences()` | Returns all the numbers `self[i]` such that `self[i] >= i+1`.
`bruhat_inversions()` | Returns the list of inversions of `self` such that the application of this inversion to `self` decrements its number of inversions.
`bruhat_inversions_iterator()` | Returns an iterator over Bruhat inversions of `self`.
`bruhat_succ()` | Returns a list of the permutations covering `self` in the Bruhat order.
`bruhat_succ_iterator()` | An iterator for the permutations covering `self` in the Bruhat order.
`bruhat_pred()` | Returns a list of the permutations covered by `self` in the Bruhat order.
`bruhat_pred_iterator()` | An iterator for the permutations covered by `self` in the Bruhat order.
`bruhat_smaller()` | Returns the combinatorial class of permutations smaller than or equal to `self` in the Bruhat order.
`bruhat_greater()` | Returns the combinatorial class of permutations greater than or equal to `self` in the Bruhat order.
`permutohedron_lequal()` | Returns `True` if `self` is less or equal to `p2` in the permutohedron order.
`permutohedron_succ()` | Returns a list of the permutations covering `self` in the permutohedron order.
`permutohedron_pred()` | Returns a list of the permutations covered by `self` in the permutohedron order.
`permutohedron_smaller()` | Returns a list of permutations smaller than or equal to `self` in the permutohedron order.
`permutohedron_greater()` | Returns a list of permutations greater than or equal to `self` in the permutohedron order.
`right_permutohedron_interval_iterator()` | Returns an iterator over permutations in an interval of the permutohedron order.
`right_permutohedron_interval()` | Returns a list of permutations in an interval of the permutohedron order.
`has_pattern()` | Tests whether the permutation `self` matches the pattern.
`avoids()` | Tests whether the permutation `self` avoids the pattern.
`pattern_positions()` | Returns the list of positions where the pattern `patt` appears in `self`.
`reverse()` | Returns the permutation obtained by reversing the 1-line notation of `self`.
`complement()` | Returns the complement of the permutation which is obtained by replacing each value \(x\) in the 1-line notation of `self` with \(n - x + 1\).
`permutation_poset()` | Returns the permutation poset of `self`.
`dict()` | Returns a dictionary corresponding to the permutation `self`.
`action()` | Returns the action of the permutation `self` on a list.
`robinson_schensted()` | Returns the pair of standard tableaux obtained by running the Robinson-Schensted Algorithm on `self`.
`left_tableau()` | Returns the left standard tableau after performing the RSK algorithm.
`right_tableau()` | Returns the right standard tableau after performing the RSK algorithm.
`increasing_tree()` | Returns the increasing tree of `self`.
`increasing_tree_shape()` | Returns the shape of the increasing tree of `self`.
`binary_search_tree()` | Returns the binary search tree of `self`.
`sylvester_class()` | Iterates over the equivalence class of `self` under sylvester congruence
`RS_partition()` | Returns the shape of the tableaux obtained by the RSK algorithm.
`remove_extra_fixed_points()` | Returns the permutation obtained by removing any fixed points at the end of `self`.
`retract_plain()` | Returns the plain retract of `self` to a smaller symmetric group \(S_m\).
`retract_direct_product()` | Returns the direct-product retract of `self` to a smaller symmetric group \(S_m\).
`retract_okounkov_vershik()` | Returns the Okounkov-Vershik retract of `self` to a smaller symmetric group \(S_m\).
`hyperoctahedral_double_coset_type()` | Returns the coset-type of `self` as a partition.
`binary_search_tree_shape()` | Returns the shape of the binary search tree of `self` (a non labelled binary tree).
`shifted_concatenation()` | Returns the right (or left) shifted concatenation of `self` with a permutation `other`.
`shifted_shuffle()` | Returns the shifted shuffle of `self` with a permutation `other`.

Other classes defined in this file

`Permutations` |
`Permutations_nk` |
`Permutations_mset` |
`Permutations_set` |
`Permutations_msetk` |
`Permutations_setk` |
`Arrangements` |
`Arrangements_msetk` |
`Arrangements_setk` |
`StandardPermutations_all` |
`StandardPermutations_n` |
`StandardPermutations_descents` |
`StandardPermutations_recoilsfiner` |
`StandardPermutations_recoilsfatter` |
`StandardPermutations_recoils` |
`StandardPermutations_bruhat_smaller` |
`StandardPermutations_bruhat_greater` |
`CyclicPermutations` |
`CyclicPermutationsOfPartition` |
`StandardPermutations_avoiding_12` |
`StandardPermutations_avoiding_21` |
`StandardPermutations_avoiding_132` |
`StandardPermutations_avoiding_123` |
`StandardPermutations_avoiding_321` |
`StandardPermutations_avoiding_231` |
`StandardPermutations_avoiding_312` |
`StandardPermutations_avoiding_213` |
`StandardPermutations_avoiding_generic` |
`PatternAvoider` |

Functions defined in this file

`from_major_code()` | Returns the permutation corresponding to major code `mc`.
`from_permutation_group_element()` | Returns a Permutation give a `PermutationGroupElement` `pge`.
`from_rank()` | Returns the permutation with the specified lexicographic rank.
`from_inversion_vector()` | Returns the permutation corresponding to inversion vector `iv`.
`from_cycles()` | Returns the permutation with given disjoint-cycle representation `cycles`.
`from_lehmer_code()` | Returns the permutation with Lehmer code `lehmer`.
`from_reduced_word()` | Returns the permutation corresponding to the reduced word `rw`.
`bistochastic_as_sum_of_permutations()` | Returns a given bistochastic matrix as a nonnegative linear combination of permutations.
`descents_composition_list()` | Returns a list of all the permutations in a given descent class (i. e., having a given descents composition).
`descents_composition_first()` | Returns the smallest element of a descent class.
`descents_composition_last()` | Returns the largest element of a descent class.
`bruhat_lequal()` | Returns `True` if `p1` is less or equal to `p2` in the Bruhat order.
`permutohedron_lequal()` | Returns `True` if `p1` is less or equal to `p2` in the permutohedron order.
`to_standard()` | Returns a standard permutation corresponding to the permutation `self`.

AUTHORS:

- Mike Hansen
- Dan Drake (2008-04-07): allow Permutation() to take lists of tuples
- Sebastien Labbe (2009-03-17): added robinson_schensted_inverse
- Travis Scrimshaw:
  - (2012-08-16): `to_standard()` no longer modifies input
  - (2013-01-19): Removed RSK implementation and moved to `rsk`.
  - (2013-07-13): Removed `CombinatorialClass` and moved permutations to the category framework.
- Darij Grinberg (2013-09-07): added methods; ameliorated trac ticket #14885 by exposing and documenting methods for global-independent multiplication.

### Classes and methods

class `sage.combinat.permutation.``Arrangements`

  Bases: `sage.combinat.permutation.Permutations`

  An arrangement of a multiset `mset` is an ordered selection without repetitions. It is represented by a list that contains only elements from `mset`, but maybe in a different order.

  `Arrangements` returns the combinatorial class of arrangements of the multiset `mset` that contain `k` elements.

  EXAMPLES:

      sage: mset = [1,1,2,3,4,4,5]
      sage: Arrangements(mset,2).list()
      [[1, 1],
       [1, 2],
       [1, 3],
       [1, 4],
       [1, 5],
       [2, 1],
       [2, 3],
       [2, 4],
       [2, 5],
       [3, 1],
       [3, 2],
       [3, 4],
       [3, 5],
       [4, 1],
       [4, 2],
       [4, 3],
       [4, 4],
       [4, 5],
       [5, 1],
       [5, 2],
       [5, 3],
       [5, 4]]
       sage: Arrangements(mset,2).cardinality()
       22
       sage: Arrangements( ["c","a","t"], 2 ).list()
       [['c', 'a'], ['c', 't'], ['a', 'c'], ['a', 't'], ['t', 'c'], ['t', 'a']]
       sage: Arrangements( ["c","a","t"], 3 ).list()
       [['c', 'a', 't'],
        ['c', 't', 'a'],
        ['a', 'c', 't'],
        ['a', 't', 'c'],
        ['t', 'c', 'a'],
        ['t', 'a', 'c']]

  `cardinality`()

    Return the cardinality of `self`.

    EXAMPLES:

        sage: A = Arrangements([1,1,2,3,4,4,5], 2)
        sage: A.cardinality()
        22

class `sage.combinat.permutation.``Arrangements_msetk`(mset, k)

  Bases: `sage.combinat.permutation.Arrangements`, `sage.combinat.permutation.Permutations_msetk`

  Arrangements of length \(k\) of a multiset \(M\).

class `sage.combinat.permutation.``Arrangements_setk`(s, k)

  Bases: `sage.combinat.permutation.Arrangements`, `sage.combinat.permutation.Permutations_setk`

  Arrangements of length \(k\) of a set \(S\).

class `sage.combinat.permutation.``CyclicPermutations`(mset)

  Bases: `sage.combinat.permutation.Permutations_mset`

  Return the class of all cyclic permutations of `mset` in cycle notation. These are the same as necklaces.

  INPUT:

  - `mset` – A multiset

  EXAMPLES:

      sage: CyclicPermutations(range(4)).list()
      [[0, 1, 2, 3],
       [0, 1, 3, 2],
       [0, 2, 1, 3],
       [0, 2, 3, 1],
       [0, 3, 1, 2],
       [0, 3, 2, 1]]
      sage: CyclicPermutations([1,1,1]).list()
      [[1, 1, 1]]

  `iterator`(distinct=False)

    EXAMPLES:

        sage: CyclicPermutations(range(4)).list() # indirect doctest
        [[0, 1, 2, 3],
         [0, 1, 3, 2],
         [0, 2, 1, 3],
         [0, 2, 3, 1],
         [0, 3, 1, 2],
         [0, 3, 2, 1]]
         sage: CyclicPermutations([1,1,1]).list()
         [[1, 1, 1]]
         sage: CyclicPermutations([1,1,1]).list(distinct=True)
         [[1, 1, 1], [1, 1, 1]]

  `list`(distinct=False)

    EXAMPLES:

        sage: CyclicPermutations(range(4)).list()
        [[0, 1, 2, 3],
         [0, 1, 3, 2],
         [0, 2, 1, 3],
         [0, 2, 3, 1],
         [0, 3, 1, 2],
         [0, 3, 2, 1]]

class `sage.combinat.permutation.``CyclicPermutationsOfPartition`(partition)

  Bases: `sage.combinat.permutation.Permutations`

  Combinations of cyclic permutations of each cell of a given partition.

  This is the same as a Cartesian product of necklaces.

  EXAMPLES:

      sage: CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]).list()
      [[[1, 2, 3, 4], [5, 6, 7]],
       [[1, 2, 4, 3], [5, 6, 7]],
       [[1, 3, 2, 4], [5, 6, 7]],
       [[1, 3, 4, 2], [5, 6, 7]],
       [[1, 4, 2, 3], [5, 6, 7]],
       [[1, 4, 3, 2], [5, 6, 7]],
       [[1, 2, 3, 4], [5, 7, 6]],
       [[1, 2, 4, 3], [5, 7, 6]],
       [[1, 3, 2, 4], [5, 7, 6]],
       [[1, 3, 4, 2], [5, 7, 6]],
       [[1, 4, 2, 3], [5, 7, 6]],
       [[1, 4, 3, 2], [5, 7, 6]]]

      sage: CyclicPermutationsOfPartition([[1,2,3,4],[4,4,4]]).list()
      [[[1, 2, 3, 4], [4, 4, 4]],
       [[1, 2, 4, 3], [4, 4, 4]],
       [[1, 3, 2, 4], [4, 4, 4]],
       [[1, 3, 4, 2], [4, 4, 4]],
       [[1, 4, 2, 3], [4, 4, 4]],
       [[1, 4, 3, 2], [4, 4, 4]]]

      sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list()
      [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]

      sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True)
      [[[1, 2, 3], [4, 4, 4]],
       [[1, 3, 2], [4, 4, 4]],
       [[1, 2, 3], [4, 4, 4]],
       [[1, 3, 2], [4, 4, 4]]]

  class `Element`

    Bases: `sage.structure.list_clone.ClonableArray`

    A cyclic permutation of a partition.

    `check`()

      Check that `self` is a valid element.

      EXAMPLES:

          sage: CP = CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]])
          sage: elt = CP[0]
          sage: elt.check()

  `CyclicPermutationsOfPartition.``iterator`(distinct=False)

    AUTHORS:

    - Robert Miller

    EXAMPLES:

        sage: CyclicPermutationsOfPartition([[1,2,3,4],[5,6,7]]).list() # indirect doctest
        [[[1, 2, 3, 4], [5, 6, 7]],
         [[1, 2, 4, 3], [5, 6, 7]],
         [[1, 3, 2, 4], [5, 6, 7]],
         [[1, 3, 4, 2], [5, 6, 7]],
         [[1, 4, 2, 3], [5, 6, 7]],
         [[1, 4, 3, 2], [5, 6, 7]],
         [[1, 2, 3, 4], [5, 7, 6]],
         [[1, 2, 4, 3], [5, 7, 6]],
         [[1, 3, 2, 4], [5, 7, 6]],
         [[1, 3, 4, 2], [5, 7, 6]],
         [[1, 4, 2, 3], [5, 7, 6]],
         [[1, 4, 3, 2], [5, 7, 6]]]

        sage: CyclicPermutationsOfPartition([[1,2,3,4],[4,4,4]]).list()
        [[[1, 2, 3, 4], [4, 4, 4]],
         [[1, 2, 4, 3], [4, 4, 4]],
         [[1, 3, 2, 4], [4, 4, 4]],
         [[1, 3, 4, 2], [4, 4, 4]],
         [[1, 4, 2, 3], [4, 4, 4]],
         [[1, 4, 3, 2], [4, 4, 4]]]

        sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list()
        [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]

        sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True)
        [[[1, 2, 3], [4, 4, 4]],
         [[1, 3, 2], [4, 4, 4]],
         [[1, 2, 3], [4, 4, 4]],
         [[1, 3, 2], [4, 4, 4]]]

  `CyclicPermutationsOfPartition.``list`(distinct=False)

    EXAMPLES:

        sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list()
        [[[1, 2, 3], [4, 4, 4]], [[1, 3, 2], [4, 4, 4]]]
        sage: CyclicPermutationsOfPartition([[1,2,3],[4,4,4]]).list(distinct=True)
        [[[1, 2, 3], [4, 4, 4]],
         [[1, 3, 2], [4, 4, 4]],
         [[1, 2, 3], [4, 4, 4]],
         [[1, 3, 2], [4, 4, 4]]]

`sage.combinat.permutation.``CyclicPermutationsOfPartition_partition`(partition)

  EXAMPLES:

      sage: sage.combinat.permutation.CyclicPermutationsOfPartition_partition([[1,2,3,4],[5,6,7]])
      doctest:...: DeprecationWarning: this class is deprecated. Use sage.combinat.permutation.CyclicPermutationsOfPartition instead
      See http://trac.sagemath.org/14772 for details.
      Cyclic permutations of partition [[1, 2, 3, 4], [5, 6, 7]]

`sage.combinat.permutation.``CyclicPermutations_mset`(partition)

  EXAMPLES:

      sage: sage.combinat.permutation.CyclicPermutations_mset(range(4))
      doctest:...: DeprecationWarning: this class is deprecated. Use sage.combinat.permutation.CyclicPermutations instead
      See http://trac.sagemath.org/14772 for details.
      Cyclic permutations of [0, 1, 2, 3]

class `sage.combinat.permutation.``PatternAvoider`(parent, patterns)

  Bases: `sage.combinat.backtrack.GenericBacktracker`

  EXAMPLES:

      sage: from sage.combinat.permutation import PatternAvoider
      sage: P = Permutations(4)
      sage: p = PatternAvoider(P, [[1,2,3]])
      sage: loads(dumps(p))
      <sage.combinat.permutation.PatternAvoider object at 0x...>

class `sage.combinat.permutation.``Permutation`(parent, l, check_input=True)

  Bases: `sage.combinat.combinat.CombinatorialObject`, `sage.structure.element.Element`

  A permutation.

  Converts `l` to a permutation on \(\{1, 2, \ldots, n\}\).

  INPUT:

  - `l` – Can be any one of the following:

    - an instance of `Permutation`,
    - list of integers, viewed as one-line permutation notation. The construction checks that you give an acceptable entry. To avoid the check, use the `check_input` option.
    - string, expressing the permutation in cycle notation.
    - list of tuples of integers, expressing the permutation in cycle notation.
    - a `PermutationGroupElement`
    - a pair of two standard tableaux of the same shape. This yields the permutation obtained from the pair using the inverse of the Robinson-Schensted algorithm.
  - `check_input` (boolean) – whether to check that input is correct. Slows

      the function down, but ensures that nothing bad happens. This is set to `True` by default.

  Warning

  Since trac ticket #13742 the input is checked for correctness : it is not accepted unless actually is a permutation on \(\{1, \ldots, n\}\). It means that some `Permutation()` objects cannot be created anymore without setting `check_input = False`, as there is no certainty that its functions can handle them, and this should be fixed in a much better way ASAP (the functions should be rewritten to handle those cases, and new tests be added).

  Warning

  There are two possible conventions for multiplying permutations, and the one currently enabled in Sage by default is the one which has \((pq)(i) = q(p(i))\) for any permutations \(p \in S_n\) and \(q \in S_n\) and any \(1 \leq i \leq n\). (This equation looks less strange when the action of permutations on numbers is written from the right: then it takes the form \(i^{pq} = (i^p)^q\), which is an associativity law). There is an alternative convention, which has \((pq)(i) = p(q(i))\) instead. The conventions can be switched at runtime using `sage.combinat.permutation.Permutations.global_options()`. It is best for code not to rely on this setting being set to a particular standard, but rather use the methods `left_action_product()` and `right_action_product()` for multiplying permutations (these methods don’t depend on the setting). See trac ticket #14885 for more details.

  Note

  The `bruhat*` methods refer to the strong Bruhat order. To use the weak Bruhat order, look under `permutohedron*`.

  EXAMPLES:

      sage: Permutation([2,1])
      [2, 1]
      sage: Permutation([2, 1, 4, 5, 3])
      [2, 1, 4, 5, 3]
      sage: Permutation('(1,2)')
      [2, 1]
      sage: Permutation('(1,2)(3,4,5)')
      [2, 1, 4, 5, 3]
      sage: Permutation( ((1,2),(3,4,5)) )
      [2, 1, 4, 5, 3]
      sage: Permutation( [(1,2),(3,4,5)] )
      [2, 1, 4, 5, 3]
      sage: Permutation( ((1,2)) )
      [2, 1]
      sage: Permutation( (1,2) )
      [2, 1]
      sage: Permutation( ((1,2),) )
      [2, 1]
      sage: Permutation( ((1,),) )
      [1]
      sage: Permutation( (1,) )
      [1]
      sage: Permutation( () )
      []
      sage: Permutation( ((),) )
      []
      sage: p = Permutation((1, 2, 5)); p
      [2, 5, 3, 4, 1]
      sage: type(p)
      <class 'sage.combinat.permutation.StandardPermutations_all_with_category.element_class'>

  Construction from a string in cycle notation:

      sage: p = Permutation( '(4,5)' ); p
      [1, 2, 3, 5, 4]

  The size of the permutation is the maximum integer appearing; add a 1-cycle to increase this:

      sage: p2 = Permutation( '(4,5)(10)' ); p2
      [1, 2, 3, 5, 4, 6, 7, 8, 9, 10]
      sage: len(p); len(p2)
      5
      10

  We construct a `Permutation` from a `PermutationGroupElement`:

      sage: g = PermutationGroupElement([2,1,3])
      sage: Permutation(g)
      [2, 1, 3]

  From a pair of tableaux of the same shape. This uses the inverse of the Robinson-Schensted algorithm:

      sage: p = [[1, 4, 7], [2, 5], [3], [6]]
      sage: q = [[1, 2, 5], [3, 6], [4], [7]]
      sage: P = Tableau(p)
      sage: Q = Tableau(q)
      sage: Permutation( (p, q) )
      [3, 6, 5, 2, 7, 4, 1]
      sage: Permutation( [p, q] )
      [3, 6, 5, 2, 7, 4, 1]
      sage: Permutation( (P, Q) )
      [3, 6, 5, 2, 7, 4, 1]
      sage: Permutation( [P, Q] )
      [3, 6, 5, 2, 7, 4, 1]

  TESTS:

      sage: Permutation([()])
      []
      sage: Permutation('()')
      []
      sage: Permutation(())
      []
      sage: Permutation( [1] )
      [1]

  From a pair of empty tableaux

      sage: Permutation( ([], []) )
      []
      sage: Permutation( [[], []] )
      []

  `_left_to_right_multiply_on_right`(rp)

    Return the permutation obtained by composing `self` with `rp` in such an order that `self` is applied first and `rp` is applied afterwards.

    This is usually denoted by either `self * rp` or `rp * self` depending on the conventions used by the author. If the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(p(i)\), then this should be denoted by `rp * self` in order to have associativity (i.e., in order to have \((p \cdot q)(i) = p(q(i))\) for all \(p\), \(q\) and \(i\)). If, on the other hand, the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(i^p\), then this should be denoted by `self * rp` in order to have associativity (i.e., in order to have \(i^{p \cdot q} = (i^p)^q\) for all \(p\), \(q\) and \(i\)).

    EXAMPLES:

        sage: p = Permutation([2,1,3])
        sage: q = Permutation([3,1,2])
        sage: p.right_action_product(q)
        [1, 3, 2]
        sage: q.right_action_product(p)
        [3, 2, 1]

  `_left_to_right_multiply_on_left`(lp)

    Return the permutation obtained by composing `self` with `lp` in such an order that `lp` is applied first and `self` is applied afterwards.

    This is usually denoted by either `self * lp` or `lp * self` depending on the conventions used by the author. If the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(p(i)\), then this should be denoted by `self * lp` in order to have associativity (i.e., in order to have \((p \cdot q)(i) = p(q(i))\) for all \(p\), \(q\) and \(i\)). If, on the other hand, the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(i^p\), then this should be denoted by `lp * self` in order to have associativity (i.e., in order to have \(i^{p \cdot q} = (i^p)^q\) for all \(p\), \(q\) and \(i\)).

    EXAMPLES:

        sage: p = Permutation([2,1,3])
        sage: q = Permutation([3,1,2])
        sage: p.left_action_product(q)
        [3, 2, 1]
        sage: q.left_action_product(p)
        [1, 3, 2]

  `RS_partition`()

    Return the shape of the tableaux obtained by applying the RSK algorithm to `self`.

    EXAMPLES:

        sage: Permutation([1,4,3,2]).RS_partition()
        [2, 1, 1]

  `action`(a)

    Return the action of the permutation `self` on a list `a`.

    The action of a permutation \(p \in S_n\) on an \(n\)-element list \((a_1, a_2, \ldots, a_n)\) is defined to be \((a_{p(1)}, a_{p(2)}, \ldots, a_{p(n)})\).

    EXAMPLES:

        sage: p = Permutation([2,1,3])
        sage: a = range(3)
        sage: p.action(a)
        [1, 0, 2]
        sage: b = [1,2,3,4]
        sage: p.action(b)
        Traceback (most recent call last):
        ...
        ValueError: len(a) must equal len(self)

        sage: q = Permutation([2,3,1])
        sage: a = range(3)
        sage: q.action(a)
        [1, 2, 0]

  `avoids`(patt)

    Test whether the permutation `self` avoids the pattern `patt`.

    EXAMPLES:

        sage: Permutation([6,2,5,4,3,1]).avoids([4,2,3,1])
        False
        sage: Permutation([6,1,2,5,4,3]).avoids([4,2,3,1])
        True
        sage: Permutation([6,1,2,5,4,3]).avoids([3,4,1,2])
        True

  `binary_search_tree`(left_to_right=True)

    Return the binary search tree associated to `self`.

    If \(w\) is a word, then the binary search tree associated to \(w\) is defined as the result of starting with an empty binary tree, and then inserting the letters of \(w\) one by one into this tree. Here, the insertion is being done according to the method `binary_search_insert()`, and the word \(w\) is being traversed from left to right.

    A permutation is regarded as a word (using one-line notation), and thus a binary search tree associated to a permutation is defined.

    If the optional keyword variable `left_to_right` is set to `False`, the word \(w\) is being traversed from right to left instead.

    EXAMPLES:

        sage: Permutation([1,4,3,2]).binary_search_tree()
        1[., 4[3[2[., .], .], .]]
        sage: Permutation([4,1,3,2]).binary_search_tree()
        4[1[., 3[2[., .], .]], .]

    By passing the option `left_to_right=False` one can have the insertion going from right to left:

        sage: Permutation([1,4,3,2]).binary_search_tree(False)
        2[1[., .], 3[., 4[., .]]]
        sage: Permutation([4,1,3,2]).binary_search_tree(False)
        2[1[., .], 3[., 4[., .]]]

    TESTS:

        sage: Permutation([]).binary_search_tree()
        .

  `binary_search_tree_shape`(left_to_right=True)

    Return the shape of the binary search tree of the permutation (a non labelled binary tree).

    EXAMPLES:

        sage: Permutation([1,4,3,2]).binary_search_tree_shape()
        [., [[[., .], .], .]]
        sage: Permutation([4,1,3,2]).binary_search_tree_shape()
        [[., [[., .], .]], .]

    By passing the option `left_to_right=False` one can have the insertion going from right to left:

        sage: Permutation([1,4,3,2]).binary_search_tree_shape(False)
        [[., .], [., [., .]]]
        sage: Permutation([4,1,3,2]).binary_search_tree_shape(False)
        [[., .], [., [., .]]]

  `bruhat_greater`()

    Returns the combinatorial class of permutations greater than or equal to `self` in the Bruhat order (on the symmetric group containing `self`).

    See `bruhat_lequal()` for the definition of the Bruhat order.

    EXAMPLES:

        sage: Permutation([4,1,2,3]).bruhat_greater().list()
        [[4, 1, 2, 3],
         [4, 1, 3, 2],
         [4, 2, 1, 3],
         [4, 2, 3, 1],
         [4, 3, 1, 2],
         [4, 3, 2, 1]]

  `bruhat_inversions`()

    Return the list of inversions of `self` such that the application of this inversion to `self` decreases its number of inversions by exactly 1.

    Equivalently, it returns the list of pairs \((i,j)\) such that \(i < j\), such that \(p(i) > p(j)\) and such that there exists no \(k\) (strictly) between \(i\) and \(j\) satisfying \(p(i) > p(k) > p(j)\).

    EXAMPLES:

        sage: Permutation([5,2,3,4,1]).bruhat_inversions()
        [[0, 1], [0, 2], [0, 3], [1, 4], [2, 4], [3, 4]]
        sage: Permutation([6,1,4,5,2,3]).bruhat_inversions()
        [[0, 1], [0, 2], [0, 3], [2, 4], [2, 5], [3, 4], [3, 5]]

  `bruhat_inversions_iterator`()

    Return the iterator for the inversions of `self` such that the application of this inversion to `self` decreases its number of inversions by exactly 1.

    EXAMPLES:

        sage: list(Permutation([5,2,3,4,1]).bruhat_inversions_iterator())
        [[0, 1], [0, 2], [0, 3], [1, 4], [2, 4], [3, 4]]
        sage: list(Permutation([6,1,4,5,2,3]).bruhat_inversions_iterator())
        [[0, 1], [0, 2], [0, 3], [2, 4], [2, 5], [3, 4], [3, 5]]

  `bruhat_lequal`(p2)

    Return `True` if `self` is less or equal to `p2` in the Bruhat order.

    The Bruhat order (also called strong Bruhat order or Chevalley order) on the symmetric group \(S_n\) is the partial order on \(S_n\) determined by the following condition: If \(p\) is a permutation, and \(i\) and \(j\) are two indices satisfying \(p(i) > p(j)\) and \(i < j\) (that is, \((i, j)\) is an inversion of \(p\) with \(i < j\)), then \(p \circ (i, j)\) (the permutation obtained by first switching \(i\) with \(j\) and then applying \(p\)) is smaller than \(p\) in the Bruhat order.

    One can show that a permutation \(p \in S_n\) is less or equal to a permutation \(q \in S_n\) in the Bruhat order if and only if for every \(i \in \{ 0, 1, \cdots , n \}\) and \(j \in \{ 1, 2, \cdots , n \}\), the number of the elements among \(p(1), p(2), \cdots, p(j)\) that are greater than \(i\) is \(\leq\) to the number of the elements among \(q(1), q(2), \cdots, q(j)\) that are greater than \(i\).

    This method assumes that `self` and `p2` are permutations of the same integer \(n\).

    EXAMPLES:

        sage: Permutation([2,4,3,1]).bruhat_lequal(Permutation([3,4,2,1]))
        True

        sage: Permutation([2,1,3]).bruhat_lequal(Permutation([2,3,1]))
        True
        sage: Permutation([2,1,3]).bruhat_lequal(Permutation([3,1,2]))
        True
        sage: Permutation([2,1,3]).bruhat_lequal(Permutation([1,2,3]))
        False
        sage: Permutation([1,3,2]).bruhat_lequal(Permutation([2,1,3]))
        False
        sage: Permutation([1,3,2]).bruhat_lequal(Permutation([2,3,1]))
        True
        sage: Permutation([2,3,1]).bruhat_lequal(Permutation([1,3,2]))
        False
        sage: sorted( [len([b for b in Permutations(3) if a.bruhat_lequal(b)])
        ....:          for a in Permutations(3)] )
        [1, 2, 2, 4, 4, 6]

        sage: Permutation([]).bruhat_lequal(Permutation([]))
        True

  `bruhat_pred`()

    Return a list of the permutations strictly smaller than `self` in the Bruhat order (on the symmetric group containing `self`) such that there is no permutation between one of those and `self`.

    See `bruhat_lequal()` for the definition of the Bruhat order.

    EXAMPLES:

        sage: Permutation([6,1,4,5,2,3]).bruhat_pred()
        [[1, 6, 4, 5, 2, 3],
         [4, 1, 6, 5, 2, 3],
         [5, 1, 4, 6, 2, 3],
         [6, 1, 2, 5, 4, 3],
         [6, 1, 3, 5, 2, 4],
         [6, 1, 4, 2, 5, 3],
         [6, 1, 4, 3, 2, 5]]

  `bruhat_pred_iterator`()

    An iterator for the permutations strictly smaller than `self` in the Bruhat order (on the symmetric group containing `self`) such that there is no permutation between one of those and `self`.

    See `bruhat_lequal()` for the definition of the Bruhat order.

    EXAMPLES:

        sage: [x for x in Permutation([6,1,4,5,2,3]).bruhat_pred_iterator()]
        [[1, 6, 4, 5, 2, 3],
         [4, 1, 6, 5, 2, 3],
         [5, 1, 4, 6, 2, 3],
         [6, 1, 2, 5, 4, 3],
         [6, 1, 3, 5, 2, 4],
         [6, 1, 4, 2, 5, 3],
         [6, 1, 4, 3, 2, 5]]

  `bruhat_smaller`()

    Return the combinatorial class of permutations smaller than or equal to `self` in the Bruhat order (on the symmetric group containing `self`).

    See `bruhat_lequal()` for the definition of the Bruhat order.

    EXAMPLES:

        sage: Permutation([4,1,2,3]).bruhat_smaller().list()
        [[1, 2, 3, 4],
         [1, 2, 4, 3],
         [1, 3, 2, 4],
         [1, 4, 2, 3],
         [2, 1, 3, 4],
         [2, 1, 4, 3],
         [3, 1, 2, 4],
         [4, 1, 2, 3]]

  `bruhat_succ`()

    Return a list of the permutations strictly greater than `self` in the Bruhat order (on the symmetric group containing `self`) such that there is no permutation between one of those and `self`.

    See `bruhat_lequal()` for the definition of the Bruhat order.

    EXAMPLES:

        sage: Permutation([6,1,4,5,2,3]).bruhat_succ()
        [[6, 4, 1, 5, 2, 3],
         [6, 2, 4, 5, 1, 3],
         [6, 1, 5, 4, 2, 3],
         [6, 1, 4, 5, 3, 2]]

  `bruhat_succ_iterator`()

    An iterator for the permutations that are strictly greater than `self` in the Bruhat order (on the symmetric group containing `self`) such that there is no permutation between one of those and `self`.

    See `bruhat_lequal()` for the definition of the Bruhat order.

    EXAMPLES:

        sage: [x for x in Permutation([6,1,4,5,2,3]).bruhat_succ_iterator()]
        [[6, 4, 1, 5, 2, 3],
         [6, 2, 4, 5, 1, 3],
         [6, 1, 5, 4, 2, 3],
         [6, 1, 4, 5, 3, 2]]

  `complement`()

    Return the complement of the permutation `self`.

    The complement of a permutation \(w \in S_n\) is defined as the permutation in \(S_n\) sending each \(i\) to \(n + 1 - w(i)\).

    EXAMPLES:

        sage: Permutation([1,2,3]).complement()
        [3, 2, 1]
        sage: Permutation([1, 3, 2]).complement()
        [3, 1, 2]

  `cycle_string`(singletons=False)

    Returns a string of the permutation in cycle notation.

    If `singletons=True`, it includes 1-cycles in the string.

    EXAMPLES:

        sage: Permutation([1,2,3]).cycle_string()
        '()'
        sage: Permutation([2,1,3]).cycle_string()
        '(1,2)'
        sage: Permutation([2,3,1]).cycle_string()
        '(1,2,3)'
        sage: Permutation([2,1,3]).cycle_string(singletons=True)
        '(1,2)(3)'

  `cycle_tuples`(singletons=True)

    Return the permutation `self` as a list of disjoint cycles.

    If `singletons=False` is given, the list does not contain the singleton cycles.

    EXAMPLES:

        sage: Permutation([2,1,3,4]).to_cycles()
        [(1, 2), (3,), (4,)]
        sage: Permutation([2,1,3,4]).to_cycles(singletons=False)
        [(1, 2)]

    The algorithm is of complexity \(O(n)\) where \(n\) is the size of the given permutation.

    TESTS:

        sage: from sage.combinat.permutation import from_cycles
        sage: for n in range(1,6):
        ....:    for p in Permutations(n):
        ....:       if from_cycles(n, p.to_cycles()) != p:
        ....:          print "There is a problem with ",p
        ....:          break
        sage: size = 10000
        sage: sample = (Permutations(size).random_element() for i in range(5))
        sage: all(from_cycles(size, p.to_cycles()) == p for p in sample)
        True

    Note: there is an alternative implementation called `_to_cycle_set` which could be slightly (10%) faster for some input (typically for permutations of size in the range [100, 10000]). You can run the following benchmarks. For small permutations:

        sage: for size in range(9): # not tested
        ....:  print size
        ....:  lp = Permutations(size).list()
        ....:  timeit('[p.to_cycles(False) for p in lp]')
        ....:  timeit('[p._to_cycles_set(False) for p in lp]')
        ....:  timeit('[p._to_cycles_list(False) for p in lp]')
        ....:  timeit('[p._to_cycles_orig(False) for p in lp]')

    and larger ones:

        sage: for size in [10, 20, 50, 75, 100, 200, 500, 1000, # not tested
        ....:       2000, 5000, 10000, 15000, 20000, 30000,
        ....:       50000, 80000, 100000]:
        ....:    print(size)
        ....:    lp = [Permutations(size).random_element() for i in range(20)]
        ....:    timeit("[p.to_cycles() for p in lp]")
        ....:    timeit("[p._to_cycles_set() for p in lp]")
        ....:    timeit("[p._to_cycles_list() for p in lp]") # not tested

  `cycle_type`()

    Return a partition of `len(self)` corresponding to the cycle type of `self`. This is a non-increasing sequence of the cycle lengths of `self`.

    EXAMPLES:

        sage: Permutation([3,1,2,4]).cycle_type()
        [3, 1]

  `descent_polynomial`()

    Return the descent polynomial of the permutation `self`.

    The descent polynomial of a permutation \(p\) is the product of all the `z[p[i]]` where `i` ranges over the descents of `p`.

    A descent of a permutation `p` is an integer `i` such that `p[i] > p[i+1]`. Here, Python’s indexing convention is used, so `p[i]` means \(p(i+1)\).

    REFERENCES:

    [GarStan1984] | A. M. Garsia, Dennis Stanton. Group actions on Stanley-Reisner rings and invariants of permutation groups. Adv. in Math. 51 (1984), 107-201. http://www.sciencedirect.com/science/article/pii/0001870884900057

    EXAMPLES:

        sage: Permutation([2,1,3]).descent_polynomial()
        z1
        sage: Permutation([4,3,2,1]).descent_polynomial()
        z1*z2^2*z3^3

    Todo

    This docstring needs to be fixed. First, the definition does not match the implementation (or the examples). Second, this doesn’t seem to be defined in [GarStan1984] (the descent monomial in their (7.23) is different).

  `descents`(final_descent=False)

    Return the list of the descents of `self`.

    A descent of a permutation `p` is an integer `i` such that `p[i] > p[i+1]`. Here, Python’s indexing convention is used, so `p[i]` means \(p(i+1)\).

    With the `final_descent` option, the last position of a non-empty permutation is also considered as a descent.

    EXAMPLES:

        sage: Permutation([3,1,2]).descents()
        [0]
        sage: Permutation([1,4,3,2]).descents()
        [1, 2]
        sage: Permutation([1,4,3,2]).descents(final_descent=True)
        [1, 2, 3]

  `descents_composition`()

    Return the descent composition of `self`.

    The descent composition of a permutation \(p \in S_n\) is defined as the composition of \(n\) whose descent set equals the descent set of \(p\). Here, the descent set of \(p\) is defined as the set of all \(i \in \{ 1, 2, \ldots, n-1 \}\) satisfying \(p(i) > p(i+1)\) (note that this differs from the output of the `descents()` method, since the latter uses Python’s indexing which starts at \(0\) instead of \(1\)). The descent set of a composition \(c = (i_1, i_2, \ldots, i_k)\) is defined as the set \(\{ i_1, i_1 + i_2, i_1 + i_2 + i_3, \ldots, i_1 + i_2 + \cdots + i_{k-1} \}\).

    EXAMPLES:

        sage: Permutation([1,3,2,4]).descents_composition()
        [2, 2]
        sage: Permutation([4,1,6,7,2,3,8,5]).descents_composition()
        [1, 3, 3, 1]
        sage: Permutation([]).descents_composition()
        []

  `dict`()

    Returns a dictionary corresponding to the permutation.

    EXAMPLES:

        sage: p = Permutation([2,1,3])
        sage: d = p.dict()
        sage: d[1]
        2
        sage: d[2]
        1
        sage: d[3]
        3

  `fixed_points`()

    Return a list of the fixed points of `self`.

    EXAMPLES:

        sage: Permutation([1,3,2,4]).fixed_points()
        [1, 4]
        sage: Permutation([1,2,3,4]).fixed_points()
        [1, 2, 3, 4]

  `foata_bijection`()

    Return the image of the permutation `self` under the Foata bijection \(\phi\).

    The bijection shows that \(\mathrm{maj}\) and \(\mathrm{inv}\) are equidistributed: if \(\phi(P) = Q\), then \(\mathrm{maj}(P) = \mathrm{inv}(Q)\).

    The Foata bijection \(\phi\) is a bijection on the set of words with no two equal letters. It can be defined by induction on the size of the word: Given a word \(w_1 w_2 \cdots w_n\), start with \(\phi(w_1) = w_1\). At the \(i\)-th step, if \(\phi(w_1 w_2 \cdots w_i) = v_1 v_2 \cdots v_i\), we define \(\phi(w_1 w_2 \cdots w_i w_{i+1})\) by placing \(w_{i+1}\) on the end of the word \(v_1 v_2 \cdots v_i\) and breaking the word up into blocks as follows. If \(w_{i+1} > v_i\), place a vertical line to the right of each \(v_k\) for which \(w_{i+1} > v_k\). Otherwise, if \(w_{i+1} < v_i\), place a vertical line to the right of each \(v_k\) for which \(w_{i+1} < v_k\). In either case, place a vertical line at the start of the word as well. Now, within each block between vertical lines, cyclically shift the entries one place to the right.

    For instance, to compute \(\phi([1,4,2,5,3])\), the sequence of words is

    - \(1\),
    - \(|1|4 \to 14\),
    - \(|14|2 \to 412\),
    - \(|4|1|2|5 \to 4125\),
    - \(|4|125|3 \to 45123\).

    So \(\phi([1,4,2,5,3]) = [4,5,1,2,3]\).

    See section 2 of [FoSc78].

    REFERENCES:

    [FoSc78] | (1, 2) Dominique Foata, Marcel-Paul Schuetzenberger. Major Index and Inversion Number of Permutations. Mathematische Nachrichten, volume 83, Issue 1, pages 143-159, 1978. http://igm.univ-mlv.fr/~berstel/Mps/Travaux/A/1978-3MajorIndexMathNachr.pdf

    EXAMPLES:

        sage: Permutation([1,2,4,3]).foata_bijection()
        [4, 1, 2, 3]
        sage: Permutation([2,5,1,3,4]).foata_bijection()
        [2, 1, 3, 5, 4]

        sage: P = Permutation([2,5,1,3,4])
        sage: P.major_index() == P.foata_bijection().number_of_inversions()
        True

        sage: all( P.major_index() == P.foata_bijection().number_of_inversions()
        ....:      for P in Permutations(4) )
        True

    The example from [FoSc78]:

        sage: Permutation([7,4,9,2,6,1,5,8,3]).foata_bijection()
        [4, 7, 2, 6, 1, 9, 5, 8, 3]

    Border cases:

        sage: Permutation([]).foata_bijection()
        []
        sage: Permutation([1]).foata_bijection()
        [1]

  `has_pattern`(patt)

    Test whether the permutation `self` contains the pattern `patt`.

    EXAMPLES:

        sage: Permutation([3,5,1,4,6,2]).has_pattern([1,3,2])
        True

  `hyperoctahedral_double_coset_type`()

    Return the coset-type of `self` as a partition.

    `self` must be a permutation of even size \(2n\). The coset-type determines the double class of the permutation, that is its image in \(H_n \backslash S_{2n} / H_n\), where \(H_n\) is the \(n\)-th hyperoctahedral group.

    The coset-type is determined as follows. Consider the perfect matching \(\{\{1,2\},\{3,4\},\dots,\{2n-1,2n\}\}\) and its image by `self`, and draw them simultaneously as edges of a graph whose vertices are labeled by \(1,2,\dots,2n\). The coset-type is the ordered sequence of the semi-lengths of the cycles of this graph (see Chapter VII of [Mcd] for more details, particularly Section VII.2).

    EXAMPLE:

        sage: Permutation([3, 4, 6, 1, 5, 7, 2, 8]).hyperoctahedral_double_coset_type()
        [3, 1]
        sage: all([p.hyperoctahedral_double_coset_type() ==
        ....:      p.inverse().hyperoctahedral_double_coset_type()
        ....:       for p in Permutations(4)])
        True
        sage: Permutation([]).hyperoctahedral_double_coset_type()
        []
        sage: Permutation([3,1,2]).hyperoctahedral_double_coset_type()
        Traceback (most recent call last):
        ...
        ValueError: [3, 1, 2] is a permutation of odd size and has no coset-type

    REFERENCES:

    [Mcd] | I. G. Macdonald. Symmetric functions and Hall polynomials. Oxford University Press, second edition, 1995.

  `idescents`(final_descent=False)

    Return a list of the idescents of `self`, that is the list of the descents of `self`‘s inverse.

    A descent of a permutation `p` is an integer `i` such that `p[i] > p[i+1]`. Here, Python’s indexing convention is used, so `p[i]` means \(p(i+1)\).

    With the `final_descent` option, the last position of a non-empty permutation is also considered as a descent.

    EXAMPLES:

        sage: Permutation([2,3,1]).idescents()
        [0]
        sage: Permutation([1,4,3,2]).idescents()
        [1, 2]
        sage: Permutation([1,4,3,2]).idescents(final_descent=True)
        [1, 2, 3]

  `idescents_signature`(final_descent=False)

    Return the list obtained as follows: Each position in `self` is mapped to \(-1\) if it is an idescent and \(1\) if it is not an idescent.

    See `idescents()` for a definition of idescents.

    With the `final_descent` option, the last position of a non-empty permutation is also considered as a descent.

    EXAMPLES:

        sage: Permutation([1,4,3,2]).idescents()
        [1, 2]
        sage: Permutation([1,4,3,2]).idescents_signature()
        [1, -1, -1, 1]

  `imajor_index`(final_descent=False)

    Return the inverse major index of the permutation `self`, which is the major index of the inverse of `self`.

    The major index of a permutation \(p\) is the sum of the descents of \(p\). Since our permutation indices are 0-based, we need to add the number of descents.

    With the `final_descent` option, the last position of a non-empty permutation is also considered as a descent.

    EXAMPLES:

        sage: Permutation([2,1,3]).imajor_index()
        1
        sage: Permutation([3,4,1,2]).imajor_index()
        2
        sage: Permutation([4,3,2,1]).imajor_index()
        6

  `increasing_tree`(compare=<built-in function min>)

    Return the increasing tree associated to `self`.

    EXAMPLES:

        sage: Permutation([1,4,3,2]).increasing_tree()
        1[., 2[3[4[., .], .], .]]
        sage: Permutation([4,1,3,2]).increasing_tree()
        1[4[., .], 2[3[., .], .]]

    By passing the option `compare=max` one can have the decreasing tree instead:

        sage: Permutation([2,3,4,1]).increasing_tree(max)
        4[3[2[., .], .], 1[., .]]
        sage: Permutation([2,3,1,4]).increasing_tree(max)
        4[3[2[., .], 1[., .]], .]

  `increasing_tree_shape`(compare=<built-in function min>)

    Return the shape of the increasing tree associated with the permutation.

    EXAMPLES:

        sage: Permutation([1,4,3,2]).increasing_tree_shape()
        [., [[[., .], .], .]]
        sage: Permutation([4,1,3,2]).increasing_tree_shape()
        [[., .], [[., .], .]]

    By passing the option `compare=max` one can have the decreasing tree instead:

        sage: Permutation([2,3,4,1]).increasing_tree_shape(max)
        [[[., .], .], [., .]]
        sage: Permutation([2,3,1,4]).increasing_tree_shape(max)
        [[[., .], [., .]], .]

  `inverse`()

    Return the inverse of `self`.

    EXAMPLES:

        sage: Permutation([3,8,5,10,9,4,6,1,7,2]).inverse()
        [8, 10, 1, 6, 3, 7, 9, 2, 5, 4]
        sage: Permutation([2, 4, 1, 5, 3]).inverse()
        [3, 1, 5, 2, 4]
        sage: ~Permutation([2, 4, 1, 5, 3])
        [3, 1, 5, 2, 4]

  `inversions`()

    Return a list of the inversions of `self`.

    An inversion of a permutation \(p\) is a pair \((i, j)\) such that \(i < j\) and \(p(i) > p(j)\).

    EXAMPLES:

        sage: Permutation([3,2,4,1,5]).inversions()
        [(1, 2), (1, 4), (2, 4), (3, 4)]

  `is_even`()

    Return `True` if the permutation `self` is even and `False` otherwise.

    EXAMPLES:

        sage: Permutation([1,2,3]).is_even()
        True
        sage: Permutation([2,1,3]).is_even()
        False

  `ishift`(i)

    Return the `i`-shift of `self`. If an `i`-shift of `self` can’t be performed, then `self` is returned.

    An \(i\)-shift can be applied when \(i\) is not inbetween \(i-1\) and \(i+1\). The \(i\)-shift moves \(i\) to the other side, and leaves the relative positions of \(i-1\) and \(i+1\) in place. All other entries of the permutations are also left in place.

    EXAMPLES:

    Here, \(2\) is to the left of both \(1\) and \(3\). A \(2\)-shift can be applied which moves the \(2\) to the right and leaves \(1\) and \(3\) in their same relative order:

        sage: Permutation([2,1,3]).ishift(2)
        [1, 3, 2]

    All entries other than \(i\), \(i-1\) and \(i+1\) are unchanged:

        sage: Permutation([2,4,1,3]).ishift(2)
        [1, 4, 3, 2]

    Since \(2\) is between \(1\) and \(3\) in `[1,2,3]`, a \(2\)-shift cannot be applied to `[1,2,3]`

        sage: Permutation([1,2,3]).ishift(2)
        [1, 2, 3]

  `iswitch`(i)

    Return the `i`-switch of `self`. If an `i`-switch of `self` can’t be performed, then `self` is returned.

    An \(i\)-switch can be applied when the subsequence of `self` formed by the entries \(i-1\), \(i\) and \(i+1\) is neither increasing nor decreasing. In this case, this subsequence is reversed (i. e., its leftmost element and its rightmost element switch places), while all other letters of `self` are kept in place.

    EXAMPLES:

    Here, \(2\) is to the left of both \(1\) and \(3\). A \(2\)-switch can be applied which moves the \(2\) to the right and switches the relative order between \(1\) and \(3\):

        sage: Permutation([2,1,3]).iswitch(2)
        [3, 1, 2]

    All entries other than \(i-1\), \(i\) and \(i+1\) are unchanged:

        sage: Permutation([2,4,1,3]).iswitch(2)
        [3, 4, 1, 2]

    Since \(2\) is between \(1\) and \(3\) in `[1,2,3]`, a \(2\)-switch cannot be applied to `[1,2,3]`

        sage: Permutation([1,2,3]).iswitch(2)
        [1, 2, 3]

  `left_action_product`(lp)

    Return the permutation obtained by composing `self` with `lp` in such an order that `lp` is applied first and `self` is applied afterwards.

    This is usually denoted by either `self * lp` or `lp * self` depending on the conventions used by the author. If the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(p(i)\), then this should be denoted by `self * lp` in order to have associativity (i.e., in order to have \((p \cdot q)(i) = p(q(i))\) for all \(p\), \(q\) and \(i\)). If, on the other hand, the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(i^p\), then this should be denoted by `lp * self` in order to have associativity (i.e., in order to have \(i^{p \cdot q} = (i^p)^q\) for all \(p\), \(q\) and \(i\)).

    EXAMPLES:

        sage: p = Permutation([2,1,3])
        sage: q = Permutation([3,1,2])
        sage: p.left_action_product(q)
        [3, 2, 1]
        sage: q.left_action_product(p)
        [1, 3, 2]

  `left_tableau`()

    Return the left standard tableau after performing the RSK algorithm on `self`.

    EXAMPLES:

        sage: Permutation([1,4,3,2]).left_tableau()
        [[1, 2], [3], [4]]

  `length`()

    Return the Coxeter length of `self`.

    The length of a permutation \(p\) is given by the number of inversions of \(p\).

    EXAMPLES:

        sage: Permutation([5, 1, 3, 4, 2]).length()
        6

  `longest_increasing_subsequence_length`()

    Return the length of the longest increasing subsequences of `self`.

    EXAMPLES:

        sage: Permutation([2,3,1,4]).longest_increasing_subsequence_length()
        3
        sage: all([i.longest_increasing_subsequence_length() == len(RSK(i)[0][0]) for i in Permutations(5)])
        True
        sage: Permutation([]).longest_increasing_subsequence_length()
        0

  `longest_increasing_subsequences`()

    Return the list of the longest increasing subsequences of `self`.

    Note

    The algorithm is not optimal.

    EXAMPLES:

        sage: Permutation([2,3,4,1]).longest_increasing_subsequences()
        [[2, 3, 4]]
        sage: Permutation([5, 7, 1, 2, 6, 4, 3]).longest_increasing_subsequences()
        [[1, 2, 6], [1, 2, 4], [1, 2, 3]]

  `major_index`(final_descent=False)

    Return the major index of `self`.

    The major index of a permutation \(p\) is the sum of the descents of \(p\). Since our permutation indices are 0-based, we need to add the number of descents.

    With the `final_descent` option, the last position of a non-empty permutation is also considered as a descent.

    EXAMPLES:

        sage: Permutation([2,1,3]).major_index()
        1
        sage: Permutation([3,4,1,2]).major_index()
        2
        sage: Permutation([4,3,2,1]).major_index()
        6

  `next`()

    Return the permutation that follows `self` in lexicographic order on the symmetric group containing `self`. If `self` is the last permutation, then `next` returns `False`.

    EXAMPLES:

        sage: p = Permutation([1, 3, 2])
        sage: p.next()
        [2, 1, 3]
        sage: p = Permutation([4,3,2,1])
        sage: p.next()
        False

    TESTS:

        sage: p = Permutation([])
        sage: p.next()
        False

  `noninversions`(k)

    Return the list of all `k`-noninversions in `self`.

    If \(k\) is an integer and \(p \in S_n\) is a permutation, then a \(k\)-noninversion in \(p\) is defined as a strictly increasing sequence \((i_1, i_2, \ldots, i_k)\) of elements of \(\{ 1, 2, \ldots, n \}\) satisfying \(p(i_1) < p(i_2) < \cdots < p(i_k)\). (In other words, a \(k\)-noninversion in \(p\) can be regarded as a \(k\)-element subset of \(\{ 1, 2, \ldots, n \}\) on which \(p\) restricts to an increasing map.)

    EXAMPLES:

        sage: p = Permutation([3, 2, 4, 1, 5])
        sage: p.noninversions(1)
        [[3], [2], [4], [1], [5]]
        sage: p.noninversions(2)
        [[3, 4], [3, 5], [2, 4], [2, 5], [4, 5], [1, 5]]
        sage: p.noninversions(3)
        [[3, 4, 5], [2, 4, 5]]
        sage: p.noninversions(4)
        []
        sage: p.noninversions(5)
        []

    TESTS:

        sage: q = Permutation([])
        sage: q.noninversions(1)
        []

  `number_of_descents`(final_descent=False)

    Return the number of descents of `self`.

    With the `final_descent` option, the last position of a non-empty permutation is also considered as a descent.

    EXAMPLES:

        sage: Permutation([1,4,3,2]).number_of_descents()
        2
        sage: Permutation([1,4,3,2]).number_of_descents(final_descent=True)
        3

  `number_of_fixed_points`()

    Return the number of fixed points of `self`.

    EXAMPLES:

        sage: Permutation([1,3,2,4]).number_of_fixed_points()
        2
        sage: Permutation([1,2,3,4]).number_of_fixed_points()
        4

  `number_of_idescents`(final_descent=False)

    Return the number of idescents of `self`.

    See `idescents()` for a definition of idescents.

    With the `final_descent` option, the last position of a non-empty permutation is also considered as a descent.

    EXAMPLES:

        sage: Permutation([1,4,3,2]).number_of_idescents()
        2
        sage: Permutation([1,4,3,2]).number_of_idescents(final_descent=True)
        3

  `number_of_inversions`()

    Return the number of inversions in `self`.

    An inversion of a permutation is a pair of elements \((i, j)\) with \(i < j\) and \(p(i) > p(j)\).

    REFERENCES:

    - http://mathworld.wolfram.com/PermutationInversion.html

    EXAMPLES:

        sage: Permutation([3, 2, 4, 1, 5]).number_of_inversions()
        4
        sage: Permutation([1, 2, 6, 4, 7, 3, 5]).number_of_inversions()
        6

  `number_of_noninversions`(k)

    Return the number of `k`-noninversions in `self`.

    If \(k\) is an integer and \(p \in S_n\) is a permutation, then a \(k\)-noninversion in \(p\) is defined as a strictly increasing sequence \((i_1, i_2, \ldots, i_k)\) of elements of \(\{ 1, 2, \ldots, n \}\) satisfying \(p(i_1) < p(i_2) < \cdots < p(i_k)\). (In other words, a \(k\)-noninversion in \(p\) can be regarded as a \(k\)-element subset of \(\{ 1, 2, \ldots, n \}\) on which \(p\) restricts to an increasing map.)

    The number of \(k\)-noninversions in \(p\) has been denoted by \(\mathrm{noninv}_k(p)\) in [RSW2011], where conjectures and results regarding this number have been stated.

    REFERENCES:

    [RSW2011] | Victor Reiner, Franco Saliola, Volkmar Welker. Spectra of Symmetrized Shuffling Operators. Arxiv 1102.2460v2.

    EXAMPLES:

        sage: p = Permutation([3, 2, 4, 1, 5])
        sage: p.number_of_noninversions(1)
        5
        sage: p.number_of_noninversions(2)
        6
        sage: p.number_of_noninversions(3)
        2
        sage: p.number_of_noninversions(4)
        0
        sage: p.number_of_noninversions(5)
        0

    The number of \(2\)-noninversions of a permutation \(p \in S_n\) is \(\binom{n}{2}\) minus its number of inversions:

        sage: b = binomial(5, 2)
        sage: all( x.number_of_noninversions(2) == b - x.number_of_inversions()
        ....:      for x in Permutations(5) )
        True

    We also check some corner cases:

        sage: all( x.number_of_noninversions(1) == 5 for x in Permutations(5) )
        True
        sage: all( x.number_of_noninversions(0) == 1 for x in Permutations(5) )
        True
        sage: Permutation([]).number_of_noninversions(1)
        0
        sage: Permutation([]).number_of_noninversions(0)
        1
        sage: Permutation([2, 1]).number_of_noninversions(3)
        0

  `number_of_peaks`()

    Return the number of peaks of the permutation `self`.

    A peak of a permutation \(p\) is an integer \(i\) such that \(p(i-1) < p(i)\) and \(p(i) > p(i+1)\).

    EXAMPLES:

        sage: Permutation([1,3,2,4,5]).number_of_peaks()
        1
        sage: Permutation([4,1,3,2,6,5]).number_of_peaks()
        2

  `number_of_recoils`()

    Return the number of recoils of the permutation `self`.

    EXAMPLES:

        sage: Permutation([1,4,3,2]).number_of_recoils()
        2

  `number_of_saliances`()

    Return the number of saliances of `self`.

    A saliance of a permutation \(p\) is an integer \(i\) such that \(p(i) > p(j)\) for all \(j > i\).

    EXAMPLES:

        sage: Permutation([2,3,1,5,4]).number_of_saliances()
        2
        sage: Permutation([5,4,3,2,1]).number_of_saliances()
        5

  `pattern_positions`(patt)

    Return the list of positions where the pattern `patt` appears in the permutation `self`.

    EXAMPLES:

        sage: Permutation([3,5,1,4,6,2]).pattern_positions([1,3,2])
        [[0, 1, 3], [2, 3, 5], [2, 4, 5]]

  `peaks`()

    Return a list of the peaks of the permutation `self`.

    A peak of a permutation \(p\) is an integer \(i\) such that \(p(i-1) < p(i)\) and \(p(i) > p(i+1)\).

    EXAMPLES:

        sage: Permutation([1,3,2,4,5]).peaks()
        [1]
        sage: Permutation([4,1,3,2,6,5]).peaks()
        [2, 4]
        sage: Permutation([]).peaks()
        []

  `permutation_poset`()

    Return the permutation poset of `self`.

    The permutation poset of a permutation \(p\) is the poset with vertices \((i, p(i))\) for \(i = 1, 2, \ldots, n\) (where \(n\) is the size of \(p\)) and order inherited from \(\ZZ \times \ZZ\).

    EXAMPLES:

        sage: Permutation([3,1,5,4,2]).permutation_poset().cover_relations()
        [[(2, 1), (5, 2)],
         [(2, 1), (3, 5)],
         [(2, 1), (4, 4)],
         [(1, 3), (3, 5)],
         [(1, 3), (4, 4)]]
        sage: Permutation([]).permutation_poset().cover_relations()
        []
        sage: Permutation([1,3,2]).permutation_poset().cover_relations()
        [[(1, 1), (2, 3)], [(1, 1), (3, 2)]]
        sage: Permutation([1,2]).permutation_poset().cover_relations()
        [[(1, 1), (2, 2)]]
        sage: P = Permutation([1,5,2,4,3])
        sage: P.permutation_poset().greene_shape() == P.RS_partition()   # This should hold for any P.
        True

  `permutohedron_greater`(side='right')

    Return a list of permutations greater than or equal to `self` in the permutohedron order.

    By default, the computations are done in the right permutohedron. If you pass the option `side='left'`, then they will be done in the left permutohedron.

    See `permutohedron_lequal()` for the definition of the permutohedron orders.

    EXAMPLES:

        sage: Permutation([4,2,1,3]).permutohedron_greater()
        [[4, 2, 1, 3], [4, 2, 3, 1], [4, 3, 2, 1]]
        sage: Permutation([4,2,1,3]).permutohedron_greater(side='left')
        [[4, 2, 1, 3], [4, 3, 1, 2], [4, 3, 2, 1]]

  `permutohedron_lequal`(p2, side='right')

    Return `True` if `self` is less or equal to `p2` in the permutohedron order.

    By default, the computations are done in the right permutohedron. If you pass the option `side='left'`, then they will be done in the left permutohedron.

    For every nonnegative integer \(n\), the right (resp. left) permutohedron order (also called the right (resp. left) weak order, or the right (resp. left) weak Bruhat order) is a partial order on the symmetric group \(S_n\). It can be defined in various ways, including the following ones:

    - Two permutations \(u\) and \(v\) in \(S_n\) satisfy \(u \leq v\) in the right (resp. left) permutohedron order if and only if the (Coxeter) length of the permutation \(v^{-1} \circ u\) (resp. of the permutation \(u \circ v^{-1}\)) equals the length of \(v\) minus the length of \(u\). Here, \(p \circ q\) means the permutation obtained by applying \(q\) first and then \(p\). (Recall that the Coxeter length of a permutation is its number of inversions.)
    - Two permutations \(u\) and \(v\) in \(S_n\) satisfy \(u \leq v\) in the right (resp. left) permutohedron order if and only if every pair \((i, j)\) of elements of \(\{ 1, 2, \cdots, n \}\) such that \(i < j\) and \(u^{-1}(i) > u^{-1}(j)\) (resp. \(u(i) > u(j)\)) also satisfies \(v^{-1}(i) > v^{-1}(j)\) (resp. \(v(i) > v(j)\)).
    - A permutation \(v \in S_n\) covers a permutation \(u \in S_n\) in the right (resp. left) permutohedron order if and only if we have \(v = u \circ (i, i + 1)\) (resp. \(v = (i, i + 1) \circ u\)) for some \(i \in \{ 1, 2, \cdots, n - 1 \}\) satisfying \(u(i) < u(i + 1)\) (resp. \(u^{-1}(i) < u^{-1}(i + 1)\)). Here, again, \(p \circ q\) means the permutation obtained by applying \(q\) first and then \(p\).

    The right and the left permutohedron order are mutually isomorphic, with the isomorphism being the map sending every permutation to its inverse. Each of these orders endows the symmetric group \(S_n\) with the structure of a graded poset (the rank function being the Coxeter length).

    Warning

    The permutohedron order is not to be mistaken for the strong Bruhat order (`bruhat_lequal()`), despite both orders being occasionally referred to as the Bruhat order.

    EXAMPLES:

        sage: p = Permutation([3,2,1,4])
        sage: p.permutohedron_lequal(Permutation([4,2,1,3]))
        False
        sage: p.permutohedron_lequal(Permutation([4,2,1,3]), side='left')
        True
        sage: p.permutohedron_lequal(p)
        True

        sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([2,3,1]))
        True
        sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([3,1,2]))
        False
        sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([1,2,3]))
        False
        sage: Permutation([1,3,2]).permutohedron_lequal(Permutation([2,1,3]))
        False
        sage: Permutation([1,3,2]).permutohedron_lequal(Permutation([2,3,1]))
        False
        sage: Permutation([2,3,1]).permutohedron_lequal(Permutation([1,3,2]))
        False
        sage: Permutation([2,1,3]).permutohedron_lequal(Permutation([2,3,1]), side='left')
        False
        sage: sorted( [len([b for b in Permutations(3) if a.permutohedron_lequal(b)])
        ....:          for a in Permutations(3)] )
        [1, 2, 2, 3, 3, 6]
        sage: sorted( [len([b for b in Permutations(3) if a.permutohedron_lequal(b, side="left")])
        ....:          for a in Permutations(3)] )
        [1, 2, 2, 3, 3, 6]

        sage: Permutation([]).permutohedron_lequal(Permutation([]))
        True

  `permutohedron_pred`(side='right')

    Return a list of the permutations strictly smaller than `self` in the permutohedron order such that there is no permutation between any of those and `self`.

    By default, the computations are done in the right permutohedron. If you pass the option `side='left'`, then they will be done in the left permutohedron.

    See `permutohedron_lequal()` for the definition of the permutohedron orders.

    EXAMPLES:

        sage: p = Permutation([4,2,1,3])
        sage: p.permutohedron_pred()
        [[2, 4, 1, 3], [4, 1, 2, 3]]
        sage: p.permutohedron_pred(side='left')
        [[4, 1, 2, 3], [3, 2, 1, 4]]

  `permutohedron_smaller`(side='right')

    Return a list of permutations smaller than or equal to `self` in the permutohedron order.

    By default, the computations are done in the right permutohedron. If you pass the option `side='left'`, then they will be done in the left permutohedron.

    See `permutohedron_lequal()` for the definition of the permutohedron orders.

    EXAMPLES:

        sage: Permutation([4,2,1,3]).permutohedron_smaller()
        [[1, 2, 3, 4],
         [1, 2, 4, 3],
         [1, 4, 2, 3],
         [2, 1, 3, 4],
         [2, 1, 4, 3],
         [2, 4, 1, 3],
         [4, 1, 2, 3],
         [4, 2, 1, 3]]

        sage: Permutation([4,2,1,3]).permutohedron_smaller(side='left')
        [[1, 2, 3, 4],
         [1, 3, 2, 4],
         [2, 1, 3, 4],
         [2, 3, 1, 4],
         [3, 1, 2, 4],
         [3, 2, 1, 4],
         [4, 1, 2, 3],
         [4, 2, 1, 3]]

  `permutohedron_succ`(side='right')

    Return a list of the permutations strictly greater than `self` in the permutohedron order such that there is no permutation between any of those and `self`.

    By default, the computations are done in the right permutohedron. If you pass the option `side='left'`, then they will be done in the left permutohedron.

    See `permutohedron_lequal()` for the definition of the permutohedron orders.

    EXAMPLES:

        sage: p = Permutation([4,2,1,3])
        sage: p.permutohedron_succ()
        [[4, 2, 3, 1]]
        sage: p.permutohedron_succ(side='left')
        [[4, 3, 1, 2]]

  `prev`()

    Return the permutation that comes directly before `self` in lexicographic order on the symmetric group containing `self`. If `self` is the first permutation, then it returns `False`.

    EXAMPLES:

        sage: p = Permutation([1,2,3])
        sage: p.prev()
        False
        sage: p = Permutation([1,3,2])
        sage: p.prev()
        [1, 2, 3]

    TESTS:

        sage: p = Permutation([])
        sage: p.prev()
        False

    Check that trac ticket #16913 is fixed:

        sage: Permutation([1,4,3,2]).prev()
        [1, 4, 2, 3]

  `rank`()

    Return the rank of `self` in the lexicographic ordering on the symmetric group to which `self` belongs.

    EXAMPLES:

        sage: Permutation([1,2,3]).rank()
        0
        sage: Permutation([1, 2, 4, 6, 3, 5]).rank()
        10
        sage: perms = Permutations(6).list()
        sage: [p.rank() for p in perms ] == range(factorial(6))
        True

  `recoils`()

    Return the list of the positions of the recoils of `self`.

    A recoil of a permutation \(p\) is an integer \(i\) such that \(i+1\) appears to the left of \(i\) in \(p\). Here, the positions are being counted starting at \(0\). (Note that it is the positions, not the recoils themselves, which are being listed.)

    EXAMPLES:

        sage: Permutation([1,4,3,2]).recoils()
        [2, 3]
        sage: Permutation([]).recoils()
        []

  `recoils_composition`()

    Return the recoils composition of `self`.

    The recoils composition of a permutation \(p \in S_n\) is the composition of \(n\) whose descent set is the set of the recoils of \(p\) (not their positions). In other words, this is the descents composition of \(p^{-1}\).

    EXAMPLES:

        sage: Permutation([1,3,2,4]).recoils_composition()
        [2, 2]
        sage: Permutation([]).recoils_composition()
        []

  `reduced_word`()

    Return a reduced word of the permutation `self`.

    See `reduced_words()` for the definition of reduced words and a way to compute them all.

    EXAMPLES:

        sage: Permutation([3,5,4,6,2,1]).reduced_word()
        [2, 1, 4, 3, 2, 4, 3, 5, 4, 5]

        Permutation([1]).reduced_word_lexmin()
        []
        Permutation([]).reduced_word_lexmin()
        []

  `reduced_word_lexmin`()

    Return a lexicographically minimal reduced word of the permutation `self`.

    See `reduced_words()` for the definition of reduced words and a way to compute them all.

    EXAMPLES:

        sage: Permutation([3,4,2,1]).reduced_word_lexmin()
        [1, 2, 1, 3, 2]

        Permutation([1]).reduced_word_lexmin()
        []
        Permutation([]).reduced_word_lexmin()
        []

  `reduced_words`()

    Return a list of the reduced words of `self`.

    The notion of a reduced word is based on the well-known fact that every permutation can be written as a product of adjacent transpositions. In more detail: If \(n\) is a nonnegative integer, we can define the transpositions \(s_i = (i, i+1) \in S_n\) for all \(i \in \{ 1, 2, \ldots, n-1 \}\), and every \(p \in S_n\) can then be written as a product \(s_{i_1} s_{i_2} \cdots s_{i_k}\) for some sequence \((i_1, i_2, \ldots, i_k)\) of elements of \(\{ 1, 2, \ldots, n-1 \}\) (here \(\{ 1, 2, \ldots, n-1 \}\) denotes the empty set when \(n \leq 1\)). Fixing a \(p\), the sequences \((i_1, i_2, \ldots, i_k)\) of smallest length satisfying \(p = s_{i_1} s_{i_2} \cdots s_{i_k}\) are called the reduced words of \(p\). (Their length is the Coxeter length of \(p\), and can be computed using `length()`.)

    Note that the product of permutations is defined here in such a way that \((pq)(i) = p(q(i))\) for all permutations \(p\) and \(q\) and each \(i \in \{ 1, 2, \ldots, n \}\) (this is the same convention as in `left_action_product()`, but not the default semantics of the \(*\) operator on permutations in Sage). Thus, for instance, \(s_2 s_1\) is the permutation obtained by first transposing \(1\) with \(2\) and then transposing \(2\) with \(3\).

    See also

    `reduced_word()`, `reduced_word_lexmin()`

    EXAMPLES:

        sage: Permutation([2,1,3]).reduced_words()
        [[1]]
        sage: Permutation([3,1,2]).reduced_words()
        [[2, 1]]
        sage: Permutation([3,2,1]).reduced_words()
        [[1, 2, 1], [2, 1, 2]]
        sage: Permutation([3,2,4,1]).reduced_words()
        [[1, 2, 3, 1], [1, 2, 1, 3], [2, 1, 2, 3]]

        Permutation([1]).reduced_words()
        [[]]
        Permutation([]).reduced_words()
        [[]]

  `remove_extra_fixed_points`()

    Return the permutation obtained by removing any fixed points at the end of `self`.

    EXAMPLES:

        sage: Permutation([2,1,3]).remove_extra_fixed_points()
        [2, 1]
        sage: Permutation([1,2,3,4]).remove_extra_fixed_points()
        [1]

    See also

    `retract_plain()`

  `retract_direct_product`(m)

    Return the direct-product retract of the permutation `self` \(\in S_n\) to \(S_m\), where \(m \leq n\). If this retract is undefined, then `None` is returned.

    If \(p \in S_n\) is a permutation, and \(m\) is a nonnegative integer less or equal to \(n\), then the direct-product retract of \(p\) to \(S_m\) is defined only if \(p([m]) = [m]\), where \([m]\) denotes the interval \(\{1, 2, \ldots, m\}\). In this case, it is defined as the permutation written \((p(1), p(2), \ldots, p(m))\) in one-line notation.

    EXAMPLES:

        sage: Permutation([4,1,2,3,5]).retract_direct_product(4)
        [4, 1, 2, 3]
        sage: Permutation([4,1,2,3,5]).retract_direct_product(3)

        sage: Permutation([1,4,2,3,6,5]).retract_direct_product(5)
        sage: Permutation([1,4,2,3,6,5]).retract_direct_product(4)
        [1, 4, 2, 3]
        sage: Permutation([1,4,2,3,6,5]).retract_direct_product(3)
        sage: Permutation([1,4,2,3,6,5]).retract_direct_product(2)
        sage: Permutation([1,4,2,3,6,5]).retract_direct_product(1)
        [1]
        sage: Permutation([1,4,2,3,6,5]).retract_direct_product(0)
        []

        sage: all( p.retract_direct_product(3) == p for p in Permutations(3) )
        True

    See also

    `retract_plain()`, `retract_okounkov_vershik()`

  `retract_okounkov_vershik`(m)

    Return the Okounkov-Vershik retract of the permutation `self` \(\in S_n\) to \(S_m\), where \(m \leq n\).

    If \(p \in S_n\) is a permutation, and \(m\) is a nonnegative integer less or equal to \(n\), then the Okounkov-Vershik retract of \(p\) to \(S_m\) is defined as the permutation in \(S_m\) which sends every \(i \in \{1, 2, \ldots, m\}\) to \(p^{k_i}(i)\), where \(k_i\) is the smallest positive integer \(k\) satisfying \(p^k(i) \leq m\).

    In other words, the Okounkov-Vershik retract of \(p\) is the permutation whose disjoint cycle decomposition is obtained by removing all letters strictly greater than \(m\) from the decomposition of \(p\) into disjoint cycles (and removing all cycles which are emptied in the process).

    When \(m = n-1\), the Okounkov-Vershik retract (as a map \(S_n \to S_{n-1}\)) is the map \(\widetilde{p}_n\) introduced in Section 7 of [OkounkovVershik2], and appears as (3.20) in [CST10]. In the general case, the Okounkov-Vershik retract of a permutation in \(S_n\) to \(S_m\) can be obtained by first taking its Okounkov-Vershik retract to \(S_{n-1}\), then that of the resulting permutation to \(S_{n-2}\), etc. until arriving in \(S_m\).

    REFERENCES:

    [OkounkovVershik2] | A. M. Vershik, A. Yu. Okounkov. A New Approach to the Representation Thoery of the Symmetric Groups. 2. http://uk.arxiv.org/abs/math/0503040v3.

    [CST10] | Tullio Ceccherini-Silberstein, Fabio Scarabotti, Filippo Tolli. Representation Theory of the Symmetric Groups: The Okounkov-Vershik Approach, Character Formulas, and Partition Algebras. CUP 2010.

    EXAMPLES:

        sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(4)
        [4, 1, 2, 3]
        sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(3)
        [3, 1, 2]
        sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(2)
        [2, 1]
        sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(1)
        [1]
        sage: Permutation([4,1,2,3,5]).retract_okounkov_vershik(0)
        []

        sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(5)
        [1, 4, 2, 3, 5]
        sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(4)
        [1, 4, 2, 3]
        sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(3)
        [1, 3, 2]
        sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(2)
        [1, 2]
        sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(1)
        [1]
        sage: Permutation([1,4,2,3,6,5]).retract_okounkov_vershik(0)
        []

        sage: Permutation([6,5,4,3,2,1]).retract_okounkov_vershik(5)
        [1, 5, 4, 3, 2]
        sage: Permutation([6,5,4,3,2,1]).retract_okounkov_vershik(4)
        [1, 2, 4, 3]

        sage: Permutation([1,5,2,6,3,7,4,8]).retract_okounkov_vershik(4)
        [1, 3, 2, 4]

        sage: all( p.retract_direct_product(3) == p for p in Permutations(3) )
        True

    See also

    `retract_plain()`, `retract_direct_product()`

  `retract_plain`(m)

    Return the plain retract of the permutation `self` in \(S_n\) to \(S_m\), where \(m \leq n\). If this retract is undefined, then `None` is returned.

    If \(p \in S_n\) is a permutation, and \(m\) is a nonnegative integer less or equal to \(n\), then the plain retract of \(p\) to \(S_m\) is defined only if every \(i > m\) satisfies \(p(i) = i\). In this case, it is defined as the permutation written \((p(1), p(2), \ldots, p(m))\) in one-line notation.

    EXAMPLES:

        sage: Permutation([4,1,2,3,5]).retract_plain(4)
        [4, 1, 2, 3]
        sage: Permutation([4,1,2,3,5]).retract_plain(3)

        sage: Permutation([1,3,2,4,5,6]).retract_plain(3)
        [1, 3, 2]
        sage: Permutation([1,3,2,4,5,6]).retract_plain(2)

        sage: Permutation([1,2,3,4,5]).retract_plain(1)
        [1]
        sage: Permutation([1,2,3,4,5]).retract_plain(0)
        []

        sage: all( p.retract_plain(3) == p for p in Permutations(3) )
        True

    See also

    `retract_direct_product()`, `retract_okounkov_vershik()`, `remove_extra_fixed_points()`

  `reverse`()

    Returns the permutation obtained by reversing the list.

    EXAMPLES:

        sage: Permutation([3,4,1,2]).reverse()
        [2, 1, 4, 3]
        sage: Permutation([1,2,3,4,5]).reverse()
        [5, 4, 3, 2, 1]

  `right_action_product`(rp)

    Return the permutation obtained by composing `self` with `rp` in such an order that `self` is applied first and `rp` is applied afterwards.

    This is usually denoted by either `self * rp` or `rp * self` depending on the conventions used by the author. If the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(p(i)\), then this should be denoted by `rp * self` in order to have associativity (i.e., in order to have \((p \cdot q)(i) = p(q(i))\) for all \(p\), \(q\) and \(i\)). If, on the other hand, the value of a permutation \(p \in S_n\) on an integer \(i \in \{ 1, 2, \cdots, n \}\) is denoted by \(i^p\), then this should be denoted by `self * rp` in order to have associativity (i.e., in order to have \(i^{p \cdot q} = (i^p)^q\) for all \(p\), \(q\) and \(i\)).

    EXAMPLES:

        sage: p = Permutation([2,1,3])
        sage: q = Permutation([3,1,2])
        sage: p.right_action_product(q)
        [1, 3, 2]
        sage: q.right_action_product(p)
        [3, 2, 1]

  `right_permutohedron_interval`(other)

    Return the list of the permutations belonging to the right permutohedron interval where `self` is the minimal element and `other` the maximal element.

    See `permutohedron_lequal()` for the definition of the permutohedron orders.

    EXAMPLES:

        sage: Permutation([2, 1, 4, 5, 3]).right_permutohedron_interval(Permutation([2, 5, 4, 1, 3]))
        [[2, 1, 4, 5, 3], [2, 1, 5, 4, 3], [2, 4, 1, 5, 3], [2, 4, 5, 1, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]]

    TESTS:

        sage: Permutation([]).right_permutohedron_interval(Permutation([]))
        [[]]
        sage: Permutation([3, 1, 2]).right_permutohedron_interval(Permutation([3, 1, 2]))
        [[3, 1, 2]]
        sage: Permutation([1, 3, 2, 4]).right_permutohedron_interval(Permutation([3, 4, 2, 1]))
        [[1, 3, 2, 4], [1, 3, 4, 2], [3, 1, 2, 4], [3, 1, 4, 2], [3, 2, 1, 4], [3, 2, 4, 1], [3, 4, 1, 2], [3, 4, 2, 1]]
        sage: Permutation([2, 1, 4, 5, 3]).right_permutohedron_interval(Permutation([2, 5, 4, 1, 3]))
        [[2, 1, 4, 5, 3], [2, 1, 5, 4, 3], [2, 4, 1, 5, 3], [2, 4, 5, 1, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]]
        sage: Permutation([2, 5, 4, 1, 3]).right_permutohedron_interval(Permutation([2, 1, 4, 5, 3]))
        Traceback (most recent call last):
        ...
        ValueError: [2, 5, 4, 1, 3] must be lower or equal than [2, 1, 4, 5, 3] for the right permutohedron order
        sage: Permutation([2, 4, 1, 3]).right_permutohedron_interval(Permutation([2, 1, 4, 5, 3]))
        Traceback (most recent call last):
        ...
        ValueError: len([2, 4, 1, 3]) and len([2, 1, 4, 5, 3]) must be equal

  `right_permutohedron_interval_iterator`(other)

    Return an iterator on the permutations (represented as integer lists) belonging to the right permutohedron interval where `self` is the minimal element and `other` the maximal element.

    See `permutohedron_lequal()` for the definition of the permutohedron orders.

    EXAMPLES:

        sage: Permutation([2, 1, 4, 5, 3]).right_permutohedron_interval(Permutation([2, 5, 4, 1, 3])) # indirect doctest
        [[2, 1, 4, 5, 3], [2, 1, 5, 4, 3], [2, 4, 1, 5, 3], [2, 4, 5, 1, 3], [2, 5, 1, 4, 3], [2, 5, 4, 1, 3]]

  `right_tableau`()

    Return the right standard tableau after performing the RSK algorithm on `self`.

    EXAMPLES:

        sage: Permutation([1,4,3,2]).right_tableau()
        [[1, 2], [3], [4]]

  `robinson_schensted`()

    Return the pair of standard tableaux obtained by running the Robinson-Schensted algorithm on `self`.

    This can also be done by running `RSK()` on `self` (with the optional argument `check_standard=True` to return standard Young tableaux).

    EXAMPLES:

        sage: Permutation([6,2,3,1,7,5,4]).robinson_schensted()
        [[[1, 3, 4], [2, 5], [6, 7]], [[1, 3, 5], [2, 6], [4, 7]]]

  `runs`()

    Return a list of the runs in the nonempty permutation `self`.

    A run in a permutation is defined to be a maximal (with respect to inclusion) nonempty increasing substring (i. e., contiguous subsequence). For instance, the runs in the permutation `[6,1,7,3,4,5,2]` are `[6]`, `[1,7]`, `[3,4,5]` and `[2]`.

    Runs in an empty permutation are not defined.

    REFERENCES:

    - http://mathworld.wolfram.com/PermutationRun.html

    EXAMPLES:

        sage: Permutation([1,2,3,4]).runs()
        [[1, 2, 3, 4]]
        sage: Permutation([4,3,2,1]).runs()
        [[4], [3], [2], [1]]
        sage: Permutation([2,4,1,3]).runs()
        [[2, 4], [1, 3]]
        sage: Permutation([1]).runs()
        [[1]]

    The example from above:

        sage: Permutation([6,1,7,3,4,5,2]).runs()
        [[6], [1, 7], [3, 4, 5], [2]]

    The number of runs in a nonempty permutation equals its number of descents plus 1:

        sage: all( len(p.runs()) == p.number_of_descents() + 1
        ....:      for p in Permutations(6) )
        True

  `saliances`()

    Return a list of the saliances of the permutation `self`.

    A saliance of a permutation \(p\) is an integer \(i\) such that \(p(i) > p(j)\) for all \(j > i\).

    EXAMPLES:

        sage: Permutation([2,3,1,5,4]).saliances()
        [3, 4]
        sage: Permutation([5,4,3,2,1]).saliances()
        [0, 1, 2, 3, 4]

  `shifted_concatenation`(other, side='right')

    Return the right (or left) shifted concatenation of `self` with a permutation `other`. These operations are also known as the Loday-Ronco over and under operations.

    INPUT:

    - `other` – a permutation, a list, a tuple, or any iterable representing a permutation.
    - `side` – (default: `"right"`) the string “left” or “right”.

    OUTPUT:

    If `side` is `"right"`, the method returns the permutation obtained by concatenating `self` with the letters of `other` incremented by the size of `self`. This is what is called `side / other` in [LodRon0102066], and denoted as the “over” operation. Otherwise, i. e., when `side` is `"left"`, the method returns the permutation obtained by concatenating the letters of `other` incremented by the size of `self` with `self`. This is what is called `side \ other` in [LodRon0102066] (which seems to use the \((\sigma \pi)(i) = \pi(\sigma(i))\) convention for the product of permutations).

    EXAMPLES:

        sage: Permutation([]).shifted_concatenation(Permutation([]), "right")
        []
        sage: Permutation([]).shifted_concatenation(Permutation([]), "left")
        []
        sage: Permutation([2, 4, 1, 3]).shifted_concatenation(Permutation([3, 1, 2]), "right")
        [2, 4, 1, 3, 7, 5, 6]
        sage: Permutation([2, 4, 1, 3]).shifted_concatenation(Permutation([3, 1, 2]), "left")
        [7, 5, 6, 2, 4, 1, 3]

    REFERENCES:

    [LodRon0102066] | (1, 2) Jean-Louis Loday and Maria O. Ronco. Order structure on the algebra of permutations and of planar binary trees. Arxiv math/0102066v1.

  `shifted_shuffle`(other)

    Return the shifted shuffle of two permutations `self` and `other`.

    INPUT:

    - `other` – a permutation, a list, a tuple, or any iterable representing a permutation.

    OUTPUT:

    The list of the permutations appearing in the shifted shuffle of the permutations `self` and `other`.

    EXAMPLES:

        sage: Permutation([]).shifted_shuffle(Permutation([]))
        [[]]
        sage: Permutation([1, 2, 3]).shifted_shuffle(Permutation([1]))
        [[1, 2, 3, 4], [1, 2, 4, 3], [1, 4, 2, 3], [4, 1, 2, 3]]
        sage: Permutation([1, 2]).shifted_shuffle(Permutation([2, 1]))
        [[1, 2, 4, 3], [1, 4, 2, 3], [1, 4, 3, 2], [4, 1, 2, 3], [4, 1, 3, 2], [4, 3, 1, 2]]
        sage: Permutation([1]).shifted_shuffle([1])
        [[1, 2], [2, 1]]
        sage: len(Permutation([3, 1, 5, 4, 2]).shifted_shuffle(Permutation([2, 1, 4, 3])))
        126

    The shifted shuffle product is associative. We can test this on an admittedly toy example:

        sage: all( all( all( sorted(flatten([abs.shifted_shuffle(c)
        ....:                                for abs in a.shifted_shuffle(b)]))
        ....:                == sorted(flatten([a.shifted_shuffle(bcs)
        ....:                                   for bcs in b.shifted_shuffle(c)]))
        ....:                for c in Permutations(2) )
        ....:           for b in Permutations(2) )
        ....:      for a in Permutations(2) )
        True

    The `shifted_shuffle` method on permutations gives the same permutations as the `shifted_shuffle` method on words (but is faster):

        sage: all( all( sorted(p1.shifted_shuffle(p2))
        ....:           == sorted([Permutation(p) for p in
        ....:                      Word(p1).shifted_shuffle(Word(p2))])
        ....:           for p2 in Permutations(3) )
        ....:      for p1 in Permutations(2) )
        True

  `show`(representation='cycles', orientation='landscape', **args)

    Display the permutation as a drawing.

    INPUT:

    - `representation` – different kinds of drawings are available

      - `"cycles"` (default) – the permutation is displayed as a collection of directed cycles

      - `"braid"` – the permutation is displayed as segments linking each element \(1, ..., n\) to its image on a parallel line.

        When using this drawing, it is also possible to display the permutation horizontally (`orientation = "landscape"`, default option) or vertically (`orientation = "portrait"`).

      - `"chord-diagram"` – the permutation is displayed as a directed graph, all of its vertices being located on a circle.

    All additional arguments are forwarded to the `show` subcalls.

    EXAMPLES:

        sage: Permutations(20).random_element().show(representation = "cycles")
        sage: Permutations(20).random_element().show(representation = "chord-diagram")
        sage: Permutations(20).random_element().show(representation = "braid")
        sage: Permutations(20).random_element().show(representation = "braid", orientation='portrait')

    TESTS:

        sage: Permutations(20).random_element().show(representation = "modern_art")
        Traceback (most recent call last):
        ...
        ValueError: The value of 'representation' must be equal to 'cycles', 'chord-diagram' or 'braid'

  `sign`()

    Return the signature of the permutation `self`. This is \((-1)^l\), where \(l\) is the number of inversions of `self`.

    Note

    `sign()` can be used as an alias for `signature()`.

    EXAMPLES:

        sage: Permutation([4, 2, 3, 1, 5]).signature()
        -1
        sage: Permutation([1,3,2,5,4]).sign()
        1
        sage: Permutation([]).sign()
        1

  `signature`()

    Return the signature of the permutation `self`. This is \((-1)^l\), where \(l\) is the number of inversions of `self`.

    Note

    `sign()` can be used as an alias for `signature()`.

    EXAMPLES:

        sage: Permutation([4, 2, 3, 1, 5]).signature()
        -1
        sage: Permutation([1,3,2,5,4]).sign()
        1
        sage: Permutation([]).sign()
        1

  `simion_schmidt`(avoid=[1, 2, 3])

    Implements the Simion-Schmidt map which sends an arbitrary permutation to a pattern avoiding permutation, where the permutation pattern is one of four length-three patterns. This method also implements the bijection between (for example) `[1,2,3]`- and `[1,3,2]`-avoiding permutations.

    INPUT:

    - `avoid` – one of the patterns `[1,2,3]`, `[1,3,2]`, `[3,1,2]`, `[3,2,1]`.

    EXAMPLES:

        sage: P=Permutations(6)
        sage: p=P([4,5,1,6,3,2])
        sage: pl= [ [1,2,3], [1,3,2], [3,1,2], [3,2,1] ]
        sage: for q in pl:
        ....:     s=p.simion_schmidt(q)
        ....:     print s, s.has_pattern(q)
        ....:
        [4, 6, 1, 5, 3, 2] False
        [4, 2, 1, 3, 5, 6] False
        [4, 5, 3, 6, 2, 1] False
        [4, 5, 1, 6, 2, 3] False

  `size`()

    Return the size of `self`.

    EXAMPLES:

        sage: Permutation([3,4,1,2,5]).size()
        5

  `sylvester_class`(left_to_right=False)

    Iterate over the equivalence class of the permutation `self` under sylvester congruence.

    Sylvester congruence is an equivalence relation on the set \(S_n\) of all permutations of \(n\). It is defined as the smallest equivalence relation such that every permutation of the form \(uacvbw\) with \(u\), \(v\) and \(w\) being words and \(a\), \(b\) and \(c\) being letters satisfying \(a \leq b < c\) is equivalent to the permutation \(ucavbw\). (Here, permutations are regarded as words by way of one-line notation.) This definition comes from [HNT05], Definition 8, where it is more generally applied to arbitrary words.

    The equivalence class of a permutation \(p \in S_n\) under sylvester congruence is called the sylvester class of \(p\). It is an interval in the right permutohedron order (see `permutohedron_lequal()`) on \(S_n\).

    This is related to the `sylvester_class()` method in that the equivalence class of a permutation \(\pi\) under sylvester congruence is the sylvester class of the right-to-left binary search tree of \(\pi\). However, the present method yields permutations, while the method on labelled binary trees yields plain lists.

    If the variable `left_to_right` is set to `True`, the method instead iterates over the equivalence class of `self` with respect to the left sylvester congruence. The left sylvester congruence is easiest to define by saying that two permutations are equivalent under it if and only if their reverses (`reverse()`) are equivalent under (standard) sylvester congruence.

    EXAMPLES:

    The sylvester class of a permutation in \(S_5\):

        sage: p = Permutation([3, 5, 1, 2, 4])
        sage: sorted(p.sylvester_class())
        [[1, 3, 2, 5, 4],
         [1, 3, 5, 2, 4],
         [1, 5, 3, 2, 4],
         [3, 1, 2, 5, 4],
         [3, 1, 5, 2, 4],
         [3, 5, 1, 2, 4],
         [5, 1, 3, 2, 4],
         [5, 3, 1, 2, 4]]

    The sylvester class of a permutation \(p\) contains \(p\):

        sage: all( p in p.sylvester_class() for p in Permutations(4) )
        True

    Small cases:

        sage: list(Permutation([]).sylvester_class())
        [[]]

        sage: list(Permutation([1]).sylvester_class())
        [[1]]

    The sylvester classes in \(S_3\):

        sage: [sorted(p.sylvester_class()) for p in Permutations(3)]
        [[[1, 2, 3]],
         [[1, 3, 2], [3, 1, 2]],
         [[2, 1, 3]],
         [[2, 3, 1]],
         [[1, 3, 2], [3, 1, 2]],
         [[3, 2, 1]]]

    The left sylvester classes in \(S_3\):

        sage: [sorted(p.sylvester_class(left_to_right=True)) for p in Permutations(3)]
        [[[1, 2, 3]],
         [[1, 3, 2]],
         [[2, 1, 3], [2, 3, 1]],
         [[2, 1, 3], [2, 3, 1]],
         [[3, 1, 2]],
         [[3, 2, 1]]]

    A left sylvester class in \(S_5\):

        sage: p = Permutation([4, 2, 1, 5, 3])
        sage: sorted(p.sylvester_class(left_to_right=True))
        [[4, 2, 1, 3, 5],
         [4, 2, 1, 5, 3],
         [4, 2, 3, 1, 5],
         [4, 2, 3, 5, 1],
         [4, 2, 5, 1, 3],
         [4, 2, 5, 3, 1],
         [4, 5, 2, 1, 3],
         [4, 5, 2, 3, 1]]

  `to_alternating_sign_matrix`()

    Return a matrix representing the permutation in the `AlternatingSignMatrix` class.

    EXAMPLES:

        sage: m = Permutation([1,2,3]).to_alternating_sign_matrix(); m
        [1 0 0]
        [0 1 0]
        [0 0 1]
        sage: parent(m)
        Alternating sign matrices of size 3

  `to_cycles`(singletons=True)

    Return the permutation `self` as a list of disjoint cycles.

    If `singletons=False` is given, the list does not contain the singleton cycles.

    EXAMPLES:

        sage: Permutation([2,1,3,4]).to_cycles()
        [(1, 2), (3,), (4,)]
        sage: Permutation([2,1,3,4]).to_cycles(singletons=False)
        [(1, 2)]

    The algorithm is of complexity \(O(n)\) where \(n\) is the size of the given permutation.

    TESTS:

        sage: from sage.combinat.permutation import from_cycles
        sage: for n in range(1,6):
        ....:    for p in Permutations(n):
        ....:       if from_cycles(n, p.to_cycles()) != p:
        ....:          print "There is a problem with ",p
        ....:          break
        sage: size = 10000
        sage: sample = (Permutations(size).random_element() for i in range(5))
        sage: all(from_cycles(size, p.to_cycles()) == p for p in sample)
        True

    Note: there is an alternative implementation called `_to_cycle_set` which could be slightly (10%) faster for some input (typically for permutations of size in the range [100, 10000]). You can run the following benchmarks. For small permutations:

        sage: for size in range(9): # not tested
        ....:  print size
        ....:  lp = Permutations(size).list()
        ....:  timeit('[p.to_cycles(False) for p in lp]')
        ....:  timeit('[p._to_cycles_set(False) for p in lp]')
        ....:  timeit('[p._to_cycles_list(False) for p in lp]')
        ....:  timeit('[p._to_cycles_orig(False) for p in lp]')

    and larger ones:

        sage: for size in [10, 20, 50, 75, 100, 200, 500, 1000, # not tested
        ....:       2000, 5000, 10000, 15000, 20000, 30000,
        ....:       50000, 80000, 100000]:
        ....:    print(size)
        ....:    lp = [Permutations(size).random_element() for i in range(20)]
        ....:    timeit("[p.to_cycles() for p in lp]")
        ....:    timeit("[p._to_cycles_set() for p in lp]")
        ....:    timeit("[p._to_cycles_list() for p in lp]") # not tested

  `to_inversion_vector`()

    Return the inversion vector of `self`.

    The inversion vector of a permutation \(p \in S_n\) is defined as the vector \((v_1, v_2, \ldots, v_n)\), where \(v_i\) is the number of elements larger than \(i\) that appear to the left of \(i\) in the permutation \(p\).

    The algorithm is of complexity \(O(n\log(n))\) where \(n\) is the size of the given permutation.

    EXAMPLES:

        sage: Permutation([5,9,1,8,2,6,4,7,3]).to_inversion_vector()
        [2, 3, 6, 4, 0, 2, 2, 1, 0]
        sage: Permutation([8,7,2,1,9,4,6,5,10,3]).to_inversion_vector()
        [3, 2, 7, 3, 4, 3, 1, 0, 0, 0]
        sage: Permutation([3,2,4,1,5]).to_inversion_vector()
        [3, 1, 0, 0, 0]

    TESTS:

        sage: from sage.combinat.permutation import from_inversion_vector
        sage: all(from_inversion_vector(p.to_inversion_vector()) == p
        ....:   for n in range(6) for p in Permutations(n))
        True

        sage: P = Permutations(1000)
        sage: sample = (P.random_element() for i in range(5))
        sage: all(from_inversion_vector(p.to_inversion_vector()) == p
        ....:   for p in sample)
        True

  `to_lehmer_cocode`()

    Return the Lehmer cocode of the permutation `self`.

    The Lehmer cocode of a permutation \(p\) is defined as the list \((c_1, c_2, \ldots, c_n)\), where \(c_i\) is the number of \(j < i\) such that \(p(j) > p(i)\).

    EXAMPLES:

        sage: p = Permutation([2,1,3])
        sage: p.to_lehmer_cocode()
        [0, 1, 0]
        sage: q = Permutation([3,1,2])
        sage: q.to_lehmer_cocode()
        [0, 1, 1]

  `to_lehmer_code`()

    Return the Lehmer code of the permutation `self`.

    The Lehmer code of a permutation \(p\) is defined as the list \([c[1],c[2],...,c[n]]\), where \(c[i]\) is the number of \(j>i\) such that \(p(j)<p(i)\).

    EXAMPLES:

        sage: p = Permutation([2,1,3])
        sage: p.to_lehmer_code()
        [1, 0, 0]
        sage: q = Permutation([3,1,2])
        sage: q.to_lehmer_code()
        [2, 0, 0]

        sage: Permutation([1]).to_lehmer_code()
        [0]
        sage: Permutation([]).to_lehmer_code()
        []

    TESTS:

        sage: from sage.combinat.permutation import from_lehmer_code
        sage: all(from_lehmer_code(p.to_lehmer_code()) == p
        ....:   for n in range(6) for p in Permutations(n))
        True

        sage: P = Permutations(1000)
        sage: sample = (P.random_element() for i in range(5))
        sage: all(from_lehmer_code(p.to_lehmer_code()) == p
        ....:   for p in sample)
        True

  `to_major_code`(final_descent=False)

    Return the major code of the permutation `self`.

    The major code of a permutation \(p\) is defined as the sequence \((m_1-m_2, m_2-m_3, \ldots, m_n)\), where \(m_i\) is the major index of the permutation obtained by erasing all letters smaller than \(i\) from \(p\).

    With the `final_descent` option, the last position of a non-empty permutation is also considered as a descent. This has an effect on the computation of major indices.

    REFERENCES:

    - Carlitz, L. q-Bernoulli and Eulerian Numbers. Trans. Amer. Math. Soc. 76 (1954) 332-350. http://www.ams.org/journals/tran/1954-076-02/S0002-9947-1954-0060538-2/
    - Skandera, M. An Eulerian Partner for Inversions. Sem. Lothar. Combin. 46 (2001) B46d. http://www.lehigh.edu/~mas906/papers/partner.ps

    EXAMPLES:

        sage: Permutation([9,3,5,7,2,1,4,6,8]).to_major_code()
        [5, 0, 1, 0, 1, 2, 0, 1, 0]
        sage: Permutation([2,8,4,3,6,7,9,5,1]).to_major_code()
        [8, 3, 3, 1, 4, 0, 1, 0, 0]

  `to_matrix`()

    Return a matrix representing the permutation.

    EXAMPLES:

        sage: Permutation([1,2,3]).to_matrix()
        [1 0 0]
        [0 1 0]
        [0 0 1]

        sage: Permutation([1,3,2]).to_matrix()
        [1 0 0]
        [0 0 1]
        [0 1 0]

    Notice that matrix multiplication corresponds to permutation multiplication only when the permutation option mult=’r2l’

        sage: PermutationOptions(mult='r2l')
        sage: p = Permutation([2,1,3])
        sage: q = Permutation([3,1,2])
        sage: (p*q).to_matrix()
        [0 0 1]
        [0 1 0]
        [1 0 0]
        sage: p.to_matrix()*q.to_matrix()
        [0 0 1]
        [0 1 0]
        [1 0 0]
        sage: PermutationOptions(mult='l2r')
        sage: (p*q).to_matrix()
        [1 0 0]
        [0 0 1]
        [0 1 0]

  `to_permutation_group_element`()

    Returns a PermutationGroupElement equal to self.

    EXAMPLES:

        sage: Permutation([2,1,4,3]).to_permutation_group_element()
        (1,2)(3,4)
        sage: Permutation([1,2,3]).to_permutation_group_element()
        ()

  `to_tableau_by_shape`(shape)

    Return a tableau of shape `shape` with the entries in `self`. The tableau is such that the reading word (i. e., the word obtained by reading the tableau row by row, starting from the top row in English notation, with each row being read from left to right) is `self`.

    EXAMPLES:

        sage: Permutation([3,4,1,2,5]).to_tableau_by_shape([3,2])
        [[1, 2, 5], [3, 4]]
        sage: Permutation([3,4,1,2,5]).to_tableau_by_shape([3,2]).reading_word_permutation()
        [3, 4, 1, 2, 5]

  `weak_excedences`()

    Return all the numbers `self[i]` such that `self[i] >= i+1`.

    EXAMPLES:

        sage: Permutation([1,4,3,2,5]).weak_excedences()
        [1, 4, 3, 5]

class `sage.combinat.permutation.``Permutations`

  Bases: `sage.structure.parent.Parent`, `sage.structure.unique_representation.UniqueRepresentation`

  Permutations.

  `Permutations(n)` returns the class of permutations of `n`, if `n` is an integer, list, set, or string.

  `Permutations(n, k)` returns the class of length-`k` partial permutations of `n` (where `n` is any of the above things); `k` must be a nonnegative integer. A length-\(k\) partial permutation of \(n\) is defined as a \(k\)-tuple of pairwise distinct elements of \(\{ 1, 2, \ldots, n \}\).

  Valid keyword arguments are: ‘descents’, ‘bruhat_smaller’, ‘bruhat_greater’, ‘recoils_finer’, ‘recoils_fatter’, ‘recoils’, and ‘avoiding’. With the exception of ‘avoiding’, you cannot specify `n` or `k` along with a keyword.

  `Permutations(descents=(list,n))` returns the class of permutations of \(n\) with descents in the positions specified by `list`. This uses the slightly nonstandard convention that the images of \(1,2,...,n\) under the permutation are regarded as positions \(0,1,...,n-1\), so for example the presence of \(1\) in `list` signifies that the permutations \(\pi\) should satisfy \(\pi(2) > \pi(3)\). Note that `list` is supposed to be a list of positions of the descents, not the descents composition. The alternative syntax `Permutations(descents=list)` is deprecated. It used to boil down `Permutations(descents=(list, max(list) + 2)`) (unless the list `list` is empty). It does not return the class of permutations with descents composition `list`.

  `Permutations(bruhat_smaller=p)` and `Permutations(bruhat_greater=p)` return the class of permutations smaller-or-equal or greater-or-equal, respectively, than the given permutation `p` in the Bruhat order. (The Bruhat order is defined in `bruhat_lequal()`. It is also referred to as the strong Bruhat order.)

  `Permutations(recoils=p)` returns the class of permutations whose recoils composition is `p`. Unlike the `descents=(list, n)` syntax, this actually takes a composition as input.

  `Permutations(recoils_fatter=p)` and `Permutations(recoils_finer=p)` return the class of permutations whose recoils composition is fatter or finer, respectively, than the given composition `p`.

  `Permutations(n, avoiding=P)` returns the class of permutations of `n` avoiding `P`. Here `P` may be a single permutation or a list of permutations; the returned class will avoid all patterns in `P`.

  EXAMPLES:

      sage: p = Permutations(3); p
      Standard permutations of 3
      sage: p.list()
      [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]

      sage: p = Permutations(3, 2); p
      Permutations of {1,...,3} of length 2
      sage: p.list()
      [[1, 2], [1, 3], [2, 1], [2, 3], [3, 1], [3, 2]]

      sage: p = Permutations(['c', 'a', 't']); p
      Permutations of the set ['c', 'a', 't']
      sage: p.list()
      [['c', 'a', 't'],
       ['c', 't', 'a'],
       ['a', 'c', 't'],
       ['a', 't', 'c'],
       ['t', 'c', 'a'],
       ['t', 'a', 'c']]

      sage: p = Permutations(['c', 'a', 't'], 2); p
      Permutations of the set ['c', 'a', 't'] of length 2
      sage: p.list()
      [['c', 'a'], ['c', 't'], ['a', 'c'], ['a', 't'], ['t', 'c'], ['t', 'a']]

      sage: p = Permutations([1,1,2]); p
      Permutations of the multi-set [1, 1, 2]
      sage: p.list()
      [[1, 1, 2], [1, 2, 1], [2, 1, 1]]

      sage: p = Permutations([1,1,2], 2); p
      Permutations of the multi-set [1, 1, 2] of length 2
      sage: p.list()
      [[1, 1], [1, 2], [2, 1]]

      sage: p = Permutations(descents=([1], 4)); p
      Standard permutations of 4 with descents [1]
      sage: p.list()
      [[1, 3, 2, 4], [1, 4, 2, 3], [2, 3, 1, 4], [2, 4, 1, 3], [3, 4, 1, 2]]

      sage: p = Permutations(bruhat_smaller=[1,3,2,4]); p
      Standard permutations that are less than or equal to [1, 3, 2, 4] in the Bruhat order
      sage: p.list()
      [[1, 2, 3, 4], [1, 3, 2, 4]]

      sage: p = Permutations(bruhat_greater=[4,2,3,1]); p
      Standard permutations that are greater than or equal to [4, 2, 3, 1] in the Bruhat order
      sage: p.list()
      [[4, 2, 3, 1], [4, 3, 2, 1]]

      sage: p = Permutations(recoils_finer=[2,1]); p
      Standard permutations whose recoils composition is finer than [2, 1]
      sage: p.list()
      [[1, 2, 3], [1, 3, 2], [3, 1, 2]]

      sage: p = Permutations(recoils_fatter=[2,1]); p
      Standard permutations whose recoils composition is fatter than [2, 1]
      sage: p.list()
      [[1, 3, 2], [3, 1, 2], [3, 2, 1]]

      sage: p = Permutations(recoils=[2,1]); p
      Standard permutations whose recoils composition is [2, 1]
      sage: p.list()
      [[1, 3, 2], [3, 1, 2]]

      sage: p = Permutations(4, avoiding=[1,3,2]); p
      Standard permutations of 4 avoiding [1, 3, 2]
      sage: p.list()
      [[4, 1, 2, 3],
       [4, 2, 1, 3],
       [4, 2, 3, 1],
       [4, 3, 1, 2],
       [4, 3, 2, 1],
       [3, 4, 1, 2],
       [3, 4, 2, 1],
       [2, 3, 4, 1],
       [3, 2, 4, 1],
       [1, 2, 3, 4],
       [2, 1, 3, 4],
       [2, 3, 1, 4],
       [3, 1, 2, 4],
       [3, 2, 1, 4]]

      sage: p = Permutations(5, avoiding=[[3,4,1,2], [4,2,3,1]]); p
      Standard permutations of 5 avoiding [[3, 4, 1, 2], [4, 2, 3, 1]]
      sage: p.cardinality()
      88
      sage: p.random_element()
      [1, 3, 5, 4, 2]

  `Element`

    alias of `Permutation`

  `global_options`(*get_value, **set_value)

    Set the global options for elements of the permutation class. The defaults are for permutations to be displayed in list notation and the multiplication done from left to right (like in GAP) – that is, \((\pi \psi)(i) = \psi(\pi(i))\) for all \(i\).

    Note

    These options have no effect on permutation group elements.

    OPTIONS:

    - `display` – (default: `list`) Specifies how the permutations should be printed
      - `cycle` – the permutations are displayed in cycle notation (i. e., as products of disjoint cycles)
      - `list` – the permutations are displayed in list notation (aka 1-line notation)
      - `reduced_expression` – alias for `reduced_word`
      - `reduced_word` – the permutations are displayed as reduced words
      - `singleton` – the permutations are displayed in cycle notation with singleton cycles shown as well
      - `word` – alias for `reduced_word`
    - `generator_name` – (default: `s`) the letter used in latexing the reduced word
    - `latex` – (default: `list`) Specifies how the permutations should be latexed
      - `cycle` – latex in cycle notation
      - `list` – latex as a list in one-line notation
      - `oneline` – alias for `list`
      - `reduced_expression` – alias for `reduced_word`
      - `reduced_word` – latex as reduced words
      - `singleton` – latex in cycle notation with singleton cycles shown as well
      - `twoline` – latex in two-line notation
      - `word` – alias for `reduced_word`
    - `latex_empty_str` – (default: `1`) The LaTeX representation of a reduced word when said word is empty
    - `mult` – (default: `l2r`) The multiplication of permutations
      - `l2r` – left to right: \((p_1 \cdot p_2)(x) = p_2(p_1(x))\)
      - `r2l` – right to left: \((p_1 \cdot p_2)(x) = p_1(p_2(x))\)

    EXAMPLES:

        sage: p213 = Permutation([2,1,3])
        sage: p312 = Permutation([3,1,2])
        sage: Permutations.global_options(mult='l2r', display='list')
        sage: Permutations.global_options['display']
        'list'
        sage: p213
        [2, 1, 3]
        sage: Permutations.global_options(display='cycle')
        sage: p213
        (1,2)
        sage: Permutations.global_options(display='singleton')
        sage: p213
        (1,2)(3)
        sage: Permutations.global_options(display='list')

        sage: Permutations.global_options['mult']
        'l2r'
        sage: p213*p312
        [1, 3, 2]
        sage: Permutations.global_options(mult='r2l')
        sage: p213*p312
        [3, 2, 1]
        sage: Permutations.global_options.reset()

    See `GlobalOptions` for more features of these options.

class `sage.combinat.permutation.``PermutationsNK`(s, k)

  Bases: `sage.combinat.permutation.Permutations_setk`

  This exists solely for unpickling `PermutationsNK` objects created with Sage <= 6.3.

class `sage.combinat.permutation.``Permutations_mset`(mset)

  Bases: `sage.combinat.permutation.Permutations`

  Permutations of a multiset \(M\).

  A permutation of a multiset \(M\) is represented by a list that contains exactly the same elements as \(M\) (with the same multiplicities), but possibly in different order. If \(M\) is a proper set there are \(|M| !\) such permutations. Otherwise, if the first element appears \(k_1\) times, the second element appears \(k_2\) times and so on, the number of permutations is \(|M|! / (k_1! k_2! \ldots)\), which is sometimes called a multinomial coefficient.

  class `Element`

    Bases: `sage.structure.list_clone.ClonableArray`

    A permutation of an arbitrary multiset.

    `check`()

      Verify that `self` is a valid permutation of the underlying multiset.

      EXAMPLES:

          sage: S = Permutations(['c','a','c'])
          sage: elt = S(['c','c','a'])
          sage: elt.check()

  `Permutations_mset.``cardinality`()

    EXAMPLES:

        sage: Permutations([1,2,2]).cardinality()
        3
        sage: Permutations([1,1,2,2,2]).cardinality()
        10

class `sage.combinat.permutation.``Permutations_msetk`(mset, k)

  Bases: `sage.combinat.permutation.Permutations_mset`

  Length-\(k\) partial permutations of a multiset.

  A length-\(k\) partial permutation of a multiset \(M\) is represented by a list of length \(k\) whose entries are elements of \(M\), appearing in the list with a multiplicity not higher than their respective multiplicity in \(M\).

class `sage.combinat.permutation.``Permutations_nk`(n, k)

  Bases: `sage.combinat.permutation.Permutations`

  Length-\(k\) partial permutations of \(\{1, 2, \ldots, n\}\).

  class `Element`

    Bases: `sage.structure.list_clone.ClonableArray`

    A length-\(k\) partial permutation of \([n]\).

    `check`()

      Verify that `self` is a valid length-\(k\) partial permutation of \([n]\).

      EXAMPLES:

          sage: S = Permutations(4, 2)
          sage: elt = S([3, 1])
          sage: elt.check()

  `Permutations_nk.``cardinality`()

    EXAMPLES:

        sage: Permutations(3,0).cardinality()
        1
        sage: Permutations(3,1).cardinality()
        3
        sage: Permutations(3,2).cardinality()
        6
        sage: Permutations(3,3).cardinality()
        6
        sage: Permutations(3,4).cardinality()
        0

  `Permutations_nk.``random_element`()

    EXAMPLES:

        sage: Permutations(3,2).random_element()
        [0, 1]

class `sage.combinat.permutation.``Permutations_set`(s)

  Bases: `sage.combinat.permutation.Permutations`

  Permutations of an arbitrary given finite set.

  Here, a “permutation of a finite set \(S\)” means a list of the elements of \(S\) in which every element of \(S\) occurs exactly once. This is not to be confused with bijections from \(S\) to \(S\), which are also often called permutations in literature.

  class `Element`

    Bases: `sage.structure.list_clone.ClonableArray`

    A permutation of an arbitrary set.

    `check`()

      Verify that `self` is a valid permutation of the underlying set.

      EXAMPLES:

          sage: S = Permutations(['c','a','t'])
          sage: elt = S(['t','c','a'])
          sage: elt.check()

  `Permutations_set.``cardinality`()

    EXAMPLES:

        sage: Permutations([1,2,3]).cardinality()
        6

  `Permutations_set.``random_element`()

    EXAMPLES:

        sage: Permutations([1,2,3]).random_element()
        [1, 2, 3]

class `sage.combinat.permutation.``Permutations_setk`(s, k)

  Bases: `sage.combinat.permutation.Permutations_set`

  Length-\(k\) partial permutations of an arbitrary given finite set.

  Here, a “length-\(k\) partial permutation of a finite set \(S\)” means a list of length \(k\) whose entries are pairwise distinct and all belong to \(S\).

  `random_element`()

    EXAMPLES:

        sage: Permutations([1,2,3],2).random_element()
        [1, 2]

class `sage.combinat.permutation.``StandardPermutations_all`

  Bases: `sage.combinat.permutation.Permutations`

  All standard permutations.

class `sage.combinat.permutation.``StandardPermutations_avoiding_12`(n)

  Bases: `sage.combinat.permutation.StandardPermutations_avoiding_generic`

  TESTS:

      sage: P = Permutations(3, avoiding=[1, 2])
      sage: TestSuite(P).run()

  `cardinality`()

    Return the cardinality of `self`.

    EXAMPLES:

        sage: P = Permutations(3, avoiding=[1, 2])
        sage: P.cardinality()
        1

class `sage.combinat.permutation.``StandardPermutations_avoiding_123`(n)

  Bases: `sage.combinat.permutation.StandardPermutations_avoiding_generic`

  TESTS:

      sage: P = Permutations(3, avoiding=[2, 1, 3])
      sage: TestSuite(P).run()

  `cardinality`()

    EXAMPLES:

        sage: Permutations(5, avoiding=[1, 2, 3]).cardinality()
        42
        sage: len( Permutations(5, avoiding=[1, 2, 3]).list() )
        42

class `sage.combinat.permutation.``StandardPermutations_avoiding_132`(n)

  Bases: `sage.combinat.permutation.StandardPermutations_avoiding_generic`

  TESTS:

      sage: P = Permutations(3, avoiding=[1, 3, 2])
      sage: TestSuite(P).run()

  `cardinality`()

    EXAMPLES:

        sage: Permutations(5, avoiding=[1, 3, 2]).cardinality()
        42
        sage: len( Permutations(5, avoiding=[1, 3, 2]).list() )
        42

class `sage.combinat.permutation.``StandardPermutations_avoiding_21`(n)

  Bases: `sage.combinat.permutation.StandardPermutations_avoiding_generic`

  TESTS:

      sage: P = Permutations(3, avoiding=[2, 1])
      sage: TestSuite(P).run()

  `cardinality`()

    Return the cardinality of `self`.

    EXAMPLES:

        sage: P = Permutations(3, avoiding=[2, 1])
        sage: P.cardinality()
        1

class `sage.combinat.permutation.``StandardPermutations_avoiding_213`(n)

  Bases: `sage.combinat.permutation.StandardPermutations_avoiding_generic`

  TESTS:

      sage: P = Permutations(3, avoiding=[2, 1, 3])
      sage: TestSuite(P).run()

  `cardinality`()

    EXAMPLES:

        sage: Permutations(5, avoiding=[2, 1, 3]).cardinality()
        42
        sage: len( Permutations(5, avoiding=[2, 1, 3]).list() )
        42

class `sage.combinat.permutation.``StandardPermutations_avoiding_231`(n)

  Bases: `sage.combinat.permutation.StandardPermutations_avoiding_generic`

  TESTS:

      sage: P = Permutations(3, avoiding=[2, 3, 1])
      sage: TestSuite(P).run()

  `cardinality`()

    EXAMPLES:

        sage: Permutations(5, avoiding=[2, 3, 1]).cardinality()
        42
        sage: len( Permutations(5, avoiding=[2, 3, 1]).list() )
        42

class `sage.combinat.permutation.``StandardPermutations_avoiding_312`(n)

  Bases: `sage.combinat.permutation.StandardPermutations_avoiding_generic`

  TESTS:

      sage: P = Permutations(3, avoiding=[3, 1, 2])
      sage: TestSuite(P).run()

  `cardinality`()

    EXAMPLES:

        sage: Permutations(5, avoiding=[3, 1, 2]).cardinality()
        42
        sage: len( Permutations(5, avoiding=[3, 1, 2]).list() )
        42

class `sage.combinat.permutation.``StandardPermutations_avoiding_321`(n)

  Bases: `sage.combinat.permutation.StandardPermutations_avoiding_generic`

  TESTS:

      sage: P = Permutations(3, avoiding=[3, 2, 1])
      sage: TestSuite(P).run()

  `cardinality`()

    EXAMPLES:

        sage: Permutations(5, avoiding=[3, 2, 1]).cardinality()
        42
        sage: len( Permutations(5, avoiding=[3, 2, 1]).list() )
        42

class `sage.combinat.permutation.``StandardPermutations_avoiding_generic`(n, a)

  Bases: `sage.combinat.permutation.StandardPermutations_n`

  Generic class for subset of permutations avoiding a particular pattern.

  `cardinality`()

    Return the cardinality of `self`.

    EXAMPLES:

        sage: P = Permutations(3, avoiding=[[2, 1, 3],[1,2,3]])
        sage: P.cardinality()
        4

class `sage.combinat.permutation.``StandardPermutations_bruhat_greater`(p)

  Bases: `sage.combinat.permutation.Permutations`

  Permutations of \(\{1, \ldots, n\}\) that are greater than or equal to a permutation \(p\) in the Bruhat order.

class `sage.combinat.permutation.``StandardPermutations_bruhat_smaller`(p)

  Bases: `sage.combinat.permutation.Permutations`

  Permutations of \(\{1, \ldots, n\}\) that are less than or equal to a permutation \(p\) in the Bruhat order.

class `sage.combinat.permutation.``StandardPermutations_descents`(d, n)

  Bases: `sage.combinat.permutation.StandardPermutations_n`

  Permutations of \(\{1, \ldots, n\}\) with a fixed set of descents.

  `cardinality`()

    Return the cardinality of `self`.

    EXAMPLES:

        sage: P = Permutations(descents=([1,0,2],5))
        sage: P.cardinality()
        4

  `first`()

    Return the first permutation with descents \(d\).

    EXAMPLES:

        sage: Permutations(descents=([1,0,4,8],12)).first()
        [3, 2, 1, 4, 6, 5, 7, 8, 10, 9, 11, 12]

  `last`()

    Return the last permutation with descents \(d\).

    EXAMPLES:

        sage: Permutations(descents=([1,0,4,8],12)).last()
        [12, 11, 8, 9, 10, 4, 5, 6, 7, 1, 2, 3]

class `sage.combinat.permutation.``StandardPermutations_n`(n)

  Bases: `sage.combinat.permutation.Permutations`

  Permutations of the set \(\{1, 2, \ldots, n\}\).

  These are also called permutations of size \(n\).

  `cardinality`()

    Return the number of permutations of size \(n\), which is \(n!\).

    EXAMPLES:

        sage: Permutations(0).cardinality()
        1
        sage: Permutations(3).cardinality()
        6
        sage: Permutations(4).cardinality()
        24

  `element_in_conjugacy_classes`(nu)

    Return a permutation with cycle type `nu`.

    If the size of `nu` is smaller than the size of permutations in `self`, then some fixed points are added.

    EXAMPLES

        sage: PP = Permutations(5)
        sage: PP.element_in_conjugacy_classes([2,2])
        [2, 1, 4, 3, 5]

  `identity`()

    Return the identity permutation of size \(n\).

    EXAMPLES:

        sage: Permutations(4).identity()
        [1, 2, 3, 4]
        sage: Permutations(0).identity()
        []

  `random_element`()

    EXAMPLES:

        sage: Permutations(4).random_element()
        [1, 2, 4, 3]

  `rank`(p)

    EXAMPLES:

        sage: SP3 = Permutations(3)
        sage: map(SP3.rank, SP3)
        [0, 1, 2, 3, 4, 5]
        sage: SP0 = Permutations(0)
        sage: map(SP0.rank, SP0)
        [0]

  `unrank`(r)

    EXAMPLES:

        sage: SP3 = Permutations(3)
        sage: l = map(SP3.unrank, range(6))
        sage: l == SP3.list()
        True
        sage: SP0 = Permutations(0)
        sage: l = map(SP0.unrank, range(1))
        sage: l == SP0.list()
        True

class `sage.combinat.permutation.``StandardPermutations_recoils`(recoils)

  Bases: `sage.combinat.permutation.Permutations`

  Permutations of \(\{1, \ldots, n\}\) with a fixed recoils composition.

class `sage.combinat.permutation.``StandardPermutations_recoilsfatter`(recoils)

  Bases: `sage.combinat.permutation.Permutations`

  TESTS:

      sage: P = Permutations(recoils_fatter=[2,2])
      sage: TestSuite(P).run()

class `sage.combinat.permutation.``StandardPermutations_recoilsfiner`(recoils)

  Bases: `sage.combinat.permutation.Permutations`

  TESTS:

      sage: P = Permutations(recoils_finer=[2,2])
      sage: TestSuite(P).run()

`sage.combinat.permutation.``bistochastic_as_sum_of_permutations`(M, check=True)

  Return the positive sum of permutations corresponding to the bistochastic matrix `M`.

  A stochastic matrix is a matrix with nonnegative real entries such that the sum of the elements of any row is equal to \(1\). A bistochastic matrix is a stochastic matrix whose transpose matrix is also stochastic ( there are conditions both on the rows and on the columns ).

  According to the Birkhoff-von Neumann Theorem, any bistochastic matrix can be written as a convex combination of permutation matrices, which also means that the polytope of bistochastic matrices is integer.

  As a non-bistochastic matrix can obviously not be written as a convex combination of permutations, this theorem is an equivalence.

  This function, given a bistochastic matrix, returns the corresponding decomposition.

  INPUT:

  - `M` – A bistochastic matrix
  - `check` (boolean) – set to `True` (default) to check that the matrix is indeed bistochastic

  OUTPUT:

  - An element of `CombinatorialFreeModule`, which is a free \(F\)-module ( where \(F\) is the ground ring of the given matrix ) whose basis is indexed by the permutations.

  Note

  - In this function, we just assume 1 to be any constant : for us a matrix \(M\) is bistochastic if there exists \(c>0\) such that \(M/c\) is bistochastic.
  - You can obtain a sequence of pairs `(permutation,coeff)`, where `permutation` is a Sage `Permutation` instance, and `coeff` its corresponding coefficient from the result of this function by applying the `list` function.
  - If you are interested in the matrix corresponding to a `Permutation` you will be glad to learn about the `Permutation.to_matrix()` method.
  - The base ring of the matrix can be anything that can be coerced to `RR`.

  See also

  - `as_sum_of_permutations()` to use this method through the `Matrix` class.

  EXAMPLES:

  We create a bistochastic matrix from a convex sum of permutations, then try to deduce the decomposition from the matrix:

      sage: from sage.combinat.permutation import bistochastic_as_sum_of_permutations
      sage: L = []
      sage: L.append((9,Permutation([4, 1, 3, 5, 2])))
      sage: L.append((6,Permutation([5, 3, 4, 1, 2])))
      sage: L.append((3,Permutation([3, 1, 4, 2, 5])))
      sage: L.append((2,Permutation([1, 4, 2, 3, 5])))
      sage: M = sum([c * p.to_matrix() for (c,p) in L])
      sage: decomp = bistochastic_as_sum_of_permutations(M)
      sage: print decomp
      2*B[[1, 4, 2, 3, 5]] + 3*B[[3, 1, 4, 2, 5]] + 9*B[[4, 1, 3, 5, 2]] + 6*B[[5, 3, 4, 1, 2]]

  An exception is raised when the matrix is not positive and bistochastic:

      sage: M = Matrix([[2,3],[2,2]])
      sage: decomp = bistochastic_as_sum_of_permutations(M)
      Traceback (most recent call last):
      ...
      ValueError: The matrix is not bistochastic

      sage: bistochastic_as_sum_of_permutations(Matrix(GF(7), 2, [2,1,1,2]))
      Traceback (most recent call last):
      ...
      ValueError: The base ring of the matrix must have a coercion map to RR

      sage: bistochastic_as_sum_of_permutations(Matrix(ZZ, 2, [2,-1,-1,2]))
      Traceback (most recent call last):
      ...
      ValueError: The matrix should have nonnegative entries

`sage.combinat.permutation.``bruhat_lequal`(p1, p2)

  Return `True` if `p1` is less than `p2` in the Bruhat order.

  Algorithm from mupad-combinat.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: permutation.bruhat_lequal([2,4,3,1],[3,4,2,1])
      True

`sage.combinat.permutation.``descents_composition_first`(dc)

  Compute the smallest element of a descent class having a descent composition `dc`.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: permutation.descents_composition_first([1,1,3,4,3])
      [3, 2, 1, 4, 6, 5, 7, 8, 10, 9, 11, 12]

`sage.combinat.permutation.``descents_composition_last`(dc)

  Return the largest element of a descent class having a descent composition `dc`.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: permutation.descents_composition_last([1,1,3,4,3])
      [12, 11, 8, 9, 10, 4, 5, 6, 7, 1, 2, 3]

`sage.combinat.permutation.``descents_composition_list`(dc)

  Return a list of all the permutations that have the descent composition `dc`.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: permutation.descents_composition_list([1,2,2])
      [[2, 1, 4, 3, 5],
       [2, 1, 5, 3, 4],
       [3, 1, 4, 2, 5],
       [3, 1, 5, 2, 4],
       [4, 1, 3, 2, 5],
       [5, 1, 3, 2, 4],
       [4, 1, 5, 2, 3],
       [5, 1, 4, 2, 3],
       [3, 2, 4, 1, 5],
       [3, 2, 5, 1, 4],
       [4, 2, 3, 1, 5],
       [5, 2, 3, 1, 4],
       [4, 2, 5, 1, 3],
       [5, 2, 4, 1, 3],
       [4, 3, 5, 1, 2],
       [5, 3, 4, 1, 2]]

`sage.combinat.permutation.``from_cycles`(n, cycles)

  Return the permutation in the \(n\)-th symmetric group whose decomposition into disjoint cycles is `cycles`.

  This function checks that its input is correct (i.e. that the cycles are disjoint and their elements integers among \(1...n\)). It raises an exception otherwise.

  Warning

  It assumes that the elements are of `int` type.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: permutation.from_cycles(4, [[1,2]])
      [2, 1, 3, 4]
      sage: permutation.from_cycles(4, [[1,2,4]])
      [2, 4, 3, 1]
      sage: permutation.from_cycles(10, [[3,1],[4,5],[6,8,9]])
      [3, 2, 1, 5, 4, 8, 7, 9, 6, 10]
      sage: permutation.from_cycles(10, ((2, 5), (6, 1, 3)))
      [3, 5, 6, 4, 2, 1, 7, 8, 9, 10]
      sage: permutation.from_cycles(4, [])
      [1, 2, 3, 4]
      sage: permutation.from_cycles(4, [[]])
      [1, 2, 3, 4]
      sage: permutation.from_cycles(0, [])
      []

  Bad input (see trac ticket #13742):

      sage: Permutation("(-12,2)(3,4)")
      Traceback (most recent call last):
      ...
      ValueError: All elements should be strictly positive integers, and I just found a negative one.
      sage: Permutation("(1,2)(2,4)")
      Traceback (most recent call last):
      ...
      ValueError: An element appears twice. It should not.
      sage: permutation.from_cycles(4, [[1,18]])
      Traceback (most recent call last):
      ...
      ValueError: You claimed that this was a permutation on 1...4 but it contains 18

`sage.combinat.permutation.``from_inversion_vector`(iv)

  Return the permutation corresponding to inversion vector `iv`.

  See \(~sage.combinat.permutation.Permutation.to_inversion_vector\) for a definition of the inversion vector of a permutation.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: permutation.from_inversion_vector([3,1,0,0,0])
      [3, 2, 4, 1, 5]
      sage: permutation.from_inversion_vector([2,3,6,4,0,2,2,1,0])
      [5, 9, 1, 8, 2, 6, 4, 7, 3]
      sage: permutation.from_inversion_vector([0])
      [1]
      sage: permutation.from_inversion_vector([])
      []

`sage.combinat.permutation.``from_lehmer_code`(lehmer)

  Return the permutation with Lehmer code `lehmer`.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: Permutation([2,1,5,4,3]).to_lehmer_code()
      [1, 0, 2, 1, 0]
      sage: permutation.from_lehmer_code(_)
      [2, 1, 5, 4, 3]

`sage.combinat.permutation.``from_major_code`(mc, final_descent=False)

  Return the permutation with major code `mc`.

  The major code of a permutation is defined in `to_major_code()`.

  Warning

  This function creates illegal permutations (i.e. `Permutation([9])`, and this is dangerous as the `Permutation()` class is only designed to handle permutations on \(1...n\). This will have to be changed when Sage permutations will be able to handle anything, but right now this should be fixed. Be careful with the results.

  Warning

  If `mc` is not a major index of a permutation, then the return value of this method can be anything. Garbage in, garbage out!

  REFERENCES:

  - Skandera, M. An Eulerian Partner for Inversions. Sem. Lothar. Combin. 46 (2001) B46d.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: permutation.from_major_code([5, 0, 1, 0, 1, 2, 0, 1, 0])
      [9, 3, 5, 7, 2, 1, 4, 6, 8]
      sage: permutation.from_major_code([8, 3, 3, 1, 4, 0, 1, 0, 0])
      [2, 8, 4, 3, 6, 7, 9, 5, 1]
      sage: Permutation([2,1,6,4,7,3,5]).to_major_code()
      [3, 2, 0, 2, 2, 0, 0]
      sage: permutation.from_major_code([3, 2, 0, 2, 2, 0, 0])
      [2, 1, 6, 4, 7, 3, 5]

  TESTS:

      sage: permutation.from_major_code([])
      []

      sage: all( permutation.from_major_code(p.to_major_code()) == p
      ....:      for p in Permutations(5) )
      True

`sage.combinat.permutation.``from_permutation_group_element`(pge)

  Return a `Permutation` given a `PermutationGroupElement` `pge`.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: pge = PermutationGroupElement([(1,2),(3,4)])
      sage: permutation.from_permutation_group_element(pge)
      [2, 1, 4, 3]

`sage.combinat.permutation.``from_rank`(n, rank)

  Return the permutation of the set \(\{1,...,n\}\) with lexicographic rank `rank`. This is the permutation whose Lehmer code is the factoradic representation of `rank`. In particular, the permutation with rank \(0\) is the identity permutation.

  The permutation is computed without iterating through all of the permutations with lower rank. This makes it efficient for large permutations.

  Note

  The variable `rank` is not checked for being in the interval from \(0\) to \(n! - 1\). When outside this interval, it acts as its residue modulo \(n!\).

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: Permutation([3, 6, 5, 4, 2, 1]).rank()
      359
      sage: [permutation.from_rank(3, i) for i in range(6)]
      [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1]]
      sage: Permutations(6)[10]
      [1, 2, 4, 6, 3, 5]
      sage: permutation.from_rank(6,10)
      [1, 2, 4, 6, 3, 5]

`sage.combinat.permutation.``from_reduced_word`(rw)

  Return the permutation corresponding to the reduced word `rw`.

  See `reduced_words()` for a definition of reduced words and the convention on the order of multiplication used.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: permutation.from_reduced_word([3,2,3,1,2,3,1])
      [3, 4, 2, 1]
      sage: permutation.from_reduced_word([])
      []

`sage.combinat.permutation.``permutohedron_lequal`(p1, p2, side='right')

  Return `True` if `p1` is less than or equal to `p2` in the permutohedron order.

  By default, the computations are done in the right permutohedron. If you pass the option `side='left'`, then they will be done in the left permutohedron.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: permutation.permutohedron_lequal(Permutation([3,2,1,4]),Permutation([4,2,1,3]))
      False
      sage: permutation.permutohedron_lequal(Permutation([3,2,1,4]),Permutation([4,2,1,3]), side='left')
      True

`sage.combinat.permutation.``to_standard`(p)

  Return a standard permutation corresponding to the list `p`.

  EXAMPLES:

      sage: import sage.combinat.permutation as permutation
      sage: permutation.to_standard([4,2,7])
      [2, 1, 3]
      sage: permutation.to_standard([1,2,3])
      [1, 2, 3]
      sage: permutation.to_standard([])
      []

  TESTS:

  Does not mutate the list:

      sage: a = [1,2,4]
      sage: permutation.to_standard(a)
      [1, 2, 3]
      sage: a
      [1, 2, 4]

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