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# Derangements

AUTHORS:

- Alasdair McAndrew (2010-05): Initial version
- Travis Scrimshaw (2013-03-30): Put derangements into category framework

class `sage.combinat.derangements.``Derangement`(parent, lst)

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

  A derangement.

  A derangement on a set \(S\) is a permutation \(\sigma\) such that \(\sigma(x) \neq x\) for all \(x \in S\), i.e. \(\sigma\) is a permutation of \(S\) with no fixed points.

  `to_permutation`()

    Return the permutation corresponding to `self`.

    EXAMPLES:

        sage: D = Derangements(4)
        sage: p = D([4,3,2,1]).to_permutation(); p
        [4, 3, 2, 1]
        sage: type(p)
        <class 'sage.combinat.permutation.StandardPermutations_all_with_category.element_class'>
        sage: D = Derangements([1, 3, 3, 4])
        sage: D[0].to_permutation()
        Traceback (most recent call last):
        ...
        ValueError: Can only convert to a permutation for derangements of [1, 2, ..., n]

class `sage.combinat.derangements.``Derangements`(x)

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

  The class of all derangements of a set or multiset.

  A derangement on a set \(S\) is a permutation \(\sigma\) such that \(\sigma(x) \neq x\) for all \(x \in S\), i.e. \(\sigma\) is a permutation of \(S\) with no fixed points.

  For an integer, or a list or string with all elements distinct, the derangements are obtained by a standard result described in [DerUB]. For a list or string with repeated elements, the derangements are formed by computing all permutations of the input and discarding all non-derangements.

  INPUT:

  - `x` – Can be an integer which corresponds to derangements of \(\{1, 2, 3, \ldots, x\}\), a list, or a string

  REFERENCES:

  [DerUB] | http://www.u-bourgogne.fr/LE2I/jl.baril/derange.pdf

  - Wikipedia article Derangement

  EXAMPLES:

      sage: D1 = Derangements([2,3,4,5])
      sage: D1.list()
      [[3, 4, 5, 2],
       [5, 4, 2, 3],
       [3, 5, 2, 4],
       [4, 5, 3, 2],
       [4, 2, 5, 3],
       [5, 2, 3, 4],
       [5, 4, 3, 2],
       [4, 5, 2, 3],
       [3, 2, 5, 4]]
      sage: D1.cardinality()
      9
      sage: D1.random_element() # random
      [4, 2, 5, 3]
      sage: D2 = Derangements([1,2,3,1,2,3])
      sage: D2.cardinality()
      10
      sage: D2.list()
      [[2, 1, 1, 3, 3, 2],
       [2, 1, 2, 3, 3, 1],
       [2, 3, 1, 2, 3, 1],
       [2, 3, 1, 3, 1, 2],
       [2, 3, 2, 3, 1, 1],
       [3, 1, 1, 2, 3, 2],
       [3, 1, 2, 2, 3, 1],
       [3, 1, 2, 3, 1, 2],
       [3, 3, 1, 2, 1, 2],
       [3, 3, 2, 2, 1, 1]]
      sage: D2.random_element() # random
      [2, 3, 1, 3, 1, 2]

  `Element`

    alias of `Derangement`

  `cardinality`()

    Counts the number of derangements of a positive integer, a list, or a string. The list or string may contain repeated elements. If an integer \(n\) is given, the the value returned is the number of derangements of \([1, 2, 3, \ldots, n]\).

    For an integer, or a list or string with all elements distinct, the value is obtained by the standard result \(D_2 = 1, D_3 = 2, D_n = (n-1) (D_{n-1} + D_{n-2})\).

    For a list or string with repeated elements, the number of derangements is computed by Macmahon’s theorem. If the numbers of repeated elements are \(a_1, a_2, \ldots, a_k\) then the number of derangements is given by the coefficient of \(x_1 x_2 \cdots x_k\) in the expansion of \(\prod_{i=0}^k (S - s_i)^{a_i}\) where \(S = x_1 + x_2 + \cdots + x_k\).

    EXAMPLES:

        sage: D = Derangements(5)
        sage: D.cardinality()
        44
        sage: D = Derangements([1,44,918,67,254])
        sage: D.cardinality()
        44
        sage: D = Derangements(['A','AT','CAT','CATS','CARTS'])
        sage: D.cardinality()
        44
        sage: D = Derangements('UNCOPYRIGHTABLE')
        sage: D.cardinality()
        481066515734
        sage: D = Derangements([1,1,2,2,3,3])
        sage: D.cardinality()
        10
        sage: D = Derangements('SATTAS')
        sage: D.cardinality()
        10
        sage: D = Derangements([1,1,2,2,2])
        sage: D.cardinality()
        0

  `random_element`()

    Produces all derangements of a positive integer, a list, or a string. The list or string may contain repeated elements. If an integer \(n\) is given, then a random derangements of \([1, 2, 3, \ldots, n]\) is returned

    For an integer, or a list or string with all elements distinct, the value is obtained by an algorithm described in [Martinez08]. For a list or string with repeated elements the derangement is formed by choosing an element at random from the list of all possible derangements.

    OUTPUT:

    A single list or string containing a derangement, or an empty list if there are no derangements.

    REFERENCES:

    [Martinez08] | http://www.siam.org/proceedings/analco/2008/anl08_022martinezc.pdf

    EXAMPLES:

        sage: D = Derangements(4)
        sage: D.random_element() # random
        [2, 3, 4, 1]
        sage: D = Derangements(['A','AT','CAT','CATS','CARTS','CARETS'])
        sage: D.random_element() # random
        ['AT', 'CARTS', 'A', 'CAT', 'CARETS', 'CATS']
        sage: D = Derangements('UNCOPYRIGHTABLE')
        sage: D.random_element() # random
        ['C', 'U', 'I', 'H', 'O', 'G', 'N', 'B', 'E', 'L', 'A', 'R', 'P', 'Y', 'T']
        sage: D = Derangements([1,1,1,1,2,2,2,2,3,3,3,3])
        sage: D.random_element() # random
        [3, 2, 2, 3, 1, 3, 1, 3, 2, 1, 1, 2]
        sage: D = Derangements('ESSENCES')
        sage: D.random_element() # random
        ['N', 'E', 'E', 'C', 'S', 'S', 'S', 'E']
        sage: D = Derangements([1,1,2,2,2])
        sage: D.random_element()
        []

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