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# Integer vectors modulo the action of a permutation group

class `sage.combinat.integer_vectors_mod_permgroup.``IntegerVectorsModPermutationGroup`

  Bases: `sage.structure.unique_representation.UniqueRepresentation`

  Returns an enumerated set containing integer vectors which are maximal in their orbit under the action of the permutation group `G` for the lexicographic order.

  In Sage, a permutation group \(G\) is viewed as a subgroup of the symmetric group \(S_n\) of degree \(n\) and \(n\) is said to be the degree of \(G\). Any integer vector \(v\) is said to be canonical if it is maximal in its orbit under the action of \(G\). \(v\) is canonical if and only if

  \[v = \max_{\text{lex order}} \{g \cdot v | g \in G \}\]

  The action of \(G\) is on position. This means for example that the simple transposition \(s_1 = (1, 2)\) swaps the first and the second entries of any integer vector \(v = [a_1, a_2, a_3, \dots , a_n]\)

  \[s_1 \cdot v = [a_2, a_1, a_3, \dots , a_n]\]

  This functions returns a parent which contains a single integer vector by orbit under the action of the permutation group \(G\). The approach chosen here is to keep the maximal integer vector for the lexicographic order in each orbit. Such maximal vector will be called canonical integer vector under the action of the permutation group \(G\).

  INPUT:

  - `G` - a permutation group
  - `sum` - (default: None) - a nonnegative integer
  - `max_part` - (default: None) - a nonnegative integer setting the maximum of entries of elements
  - `sgs` - (default: None) - a strong generating system of the group \(G\). If you do not provide it, it will be calculated at the creation of the parent

  OUTPUT:

  - If `sum` and `max_part` are None, it returns the infinite enumerated set of all integer vectors (list of integers) maximal in their orbit for the lexicographic order.
  - If `sum` is an integer, it returns a finite enumerated set containing all integer vectors maximal in their orbit for the lexicographic order and whose entries sum to `sum`.

  EXAMPLES:

  Here is the set enumerating integer vectors modulo the action of the cyclic group of \(3\) elements:

      sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3)]]))
      sage: I.category()
      Join of Category of infinite enumerated sets and Category of quotients of sets
      sage: I.cardinality()
      +Infinity
      sage: I.list()
      Traceback (most recent call last):
      ...
      NotImplementedError: infinite list
      sage: p = iter(I)
      sage: for i in range(10): p.next()
      [0, 0, 0]
      [1, 0, 0]
      [2, 0, 0]
      [1, 1, 0]
      [3, 0, 0]
      [2, 1, 0]
      [2, 0, 1]
      [1, 1, 1]
      [4, 0, 0]
      [3, 1, 0]

  The method `is_canonical()` tests if any integer vector is maximal in its orbit. This method is also used in the containment test:

      sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
      sage: I.is_canonical([5,2,0,4])
      True
      sage: I.is_canonical([5,0,6,4])
      False
      sage: I.is_canonical([1,1,1,1])
      True
      sage: [2,3,1,0] in I
      False
      sage: [5,0,5,0] in I
      True
      sage: 'Bla' in I
      False
      sage: I.is_canonical('bla')
      Traceback (most recent call last):
      ...
      AssertionError: bla should be a list or a integer vector

  If you give a value to the extra argument `sum`, the set returned will be a finite set containing only canonical vectors whose entries sum to `sum`.:

      sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3)]]), sum=6)
      sage: I.cardinality()
      10
      sage: I.list()
      [[6, 0, 0], [5, 1, 0], [5, 0, 1], [4, 2, 0], [4, 1, 1],
       [4, 0, 2], [3, 3, 0], [3, 2, 1], [3, 1, 2], [2, 2, 2]]
      sage: I.category()
      Join of Category of finite enumerated sets and Category of subquotients of finite sets and Category of quotients of sets

  To get the orbit of any integer vector \(v\) under the action of the group, use the method `orbit()`; we convert the returned set of vectors into a list in increasing lexicographic order, to get a reproducible test:

      sage: sorted(I.orbit([6,0,0]))
      [[0, 0, 6], [0, 6, 0], [6, 0, 0]]
      sage: sorted(I.orbit([5,1,0]))
      [[0, 5, 1], [1, 0, 5], [5, 1, 0]]
      sage: I.orbit([2,2,2])
      {[2, 2, 2]}

  TESTS:

  Let us check that canonical integer vectors of the symmetric group are just sorted list of integers:

      sage: I = IntegerVectorsModPermutationGroup(SymmetricGroup(5)) # long time
      sage: p = iter(I) # long time
      sage: for i in range(100): # long time
      ...       v = list(p.next())
      ...       assert sorted(v, reverse=True) == v

  We now check that there is as much of canonical vectors under the symmetric group \(S_n\) whose entries sum to \(d\) than partitions of \(d\) of at most \(n\) parts:

      sage: I = IntegerVectorsModPermutationGroup(SymmetricGroup(5)) # long time
      sage: for i in range(10): # long time
      ...       d1 = I.subset(i).cardinality()
      ...       d2 = Partitions(i, max_length=5).cardinality()
      ...       print d1
      ...       assert d1 == d2
      1
      1
      2
      3
      5
      7
      10
      13
      18
      23

  We present a last corner case: trivial groups. For the trivial group `G` acting on a list of length \(n\), all integer vectors of length \(n\) are canonical:

      sage: G = PermutationGroup([[(6,)]]) # long time
      sage: G.cardinality() # long time
      1
      sage: I = IntegerVectorsModPermutationGroup(G) # long time
      sage: for i in range(10): # long time
      ...       d1 = I.subset(i).cardinality()
      ...       d2 = IntegerVectors(i,6).cardinality()
      ...       print d1
      ...       assert d1 == d2
      1
      6
      21
      56
      126
      252
      462
      792
      1287
      2002

class `sage.combinat.integer_vectors_mod_permgroup.``IntegerVectorsModPermutationGroup_All`(G, sgs=None)

  Bases: `sage.structure.unique_representation.UniqueRepresentation`, `sage.combinat.backtrack.SearchForest`

  A class for integer vectors enumerated up to the action of a permutation group.

  A Sage permutation group is viewed as a subgroup of the symmetric group \(S_n\) for a certain \(n\). This group has a natural action by position on vectors of length \(n\). This class implements a set which keeps a single vector for each orbit. We say that a vector is canonical if it is the maximum in its orbit under the action of the permutation group for the lexicographic order.

  EXAMPLES:

      sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
      sage: I
      Integer vectors of length 4 enumerated up to the action of Permutation Group with generators [(1,2,3,4)]
      sage: I.cardinality()
      +Infinity
      sage: TestSuite(I).run()
      sage: it = iter(I)
      sage: [it.next(), it.next(), it.next(), it.next(), it.next()]
      [[0, 0, 0, 0],
       [1, 0, 0, 0],
       [2, 0, 0, 0],
       [1, 1, 0, 0],
       [1, 0, 1, 0]]
      sage: x = it.next(); x
      [3, 0, 0, 0]
      sage: I.first()
      [0, 0, 0, 0]

  TESTS:

      sage: TestSuite(I).run()

  class `Element`

    Bases: `sage.structure.list_clone.ClonableIntArray`

    Element class for the set of integer vectors of given sum enumerated modulo the action of a permutation group. These vector are clonable lists of integers which must check conditions comming form the parent appearing in the method `check()`.

    TESTS:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: v = I.element_class(I, [4,3,2,1]); v
        [4, 3, 2, 1]
        sage: TestSuite(v).run()
        sage: I.element_class(I, [4,3,2,5])
        Traceback (most recent call last):
        ...
        AssertionError

    `check`()

      Checks that `self` verify the invariants needed for living in `self.parent()`.

      EXAMPLES:

          sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
          sage: v = I.an_element()
          sage: v.check()
          sage: w = I([0,4,0,0], check=False); w
          [0, 4, 0, 0]
          sage: w.check()
          Traceback (most recent call last):
          ...
          AssertionError

  `IntegerVectorsModPermutationGroup_All.``ambient`()

    Return the ambient space from which `self` is a quotient.

    EXAMPLES:

        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: S.ambient()
        Integer vectors

  `IntegerVectorsModPermutationGroup_All.``children`(x)

    Returns the list of children of the element `x`. This method is required to build the tree structure of `self` which inherits from the class `SearchForest`.

    EXAMPLES:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: I.children(I([2,1,0,0], check=False))
        [[2, 2, 0, 0], [2, 1, 1, 0], [2, 1, 0, 1]]

  `IntegerVectorsModPermutationGroup_All.``is_canonical`(v, check=True)

    Returns `True` if the integer list `v` is maximal in its orbit under the action of the permutation group given to define `self`. Such integer vectors are said to be canonical. A vector \(v\) is canonical if and only if

    \[v = \max_{\text{lex order}} \{g \cdot v | g \in G \}\]

    EXAMPLES:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: I.is_canonical([4,3,2,1])
        True
        sage: I.is_canonical([4,0,0,1])
        True
        sage: I.is_canonical([4,0,3,3])
        True
        sage: I.is_canonical([4,0,4,4])
        False

  `IntegerVectorsModPermutationGroup_All.``lift`(elt)

    Lift the element `elt` inside the ambient space from which `self` is a quotient.

    EXAMPLES:

        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: v = S.lift(S([4,3,0,1])); v
        [4, 3, 0, 1]
        sage: type(v)
        <type 'list'>

  `IntegerVectorsModPermutationGroup_All.``orbit`(v)

    Returns the orbit of the integer vector `v` under the action of the permutation group defining `self`. The result is a set.

    EXAMPLES:

    In order to get reproducible doctests, we convert the returned sets into lists in increasing lexicographic order:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: sorted(I.orbit([2,2,0,0]))
        [[0, 0, 2, 2], [0, 2, 2, 0], [2, 0, 0, 2], [2, 2, 0, 0]]
        sage: sorted(I.orbit([2,1,0,0]))
        [[0, 0, 2, 1], [0, 2, 1, 0], [1, 0, 0, 2], [2, 1, 0, 0]]
        sage: sorted(I.orbit([2,0,1,0]))
        [[0, 1, 0, 2], [0, 2, 0, 1], [1, 0, 2, 0], [2, 0, 1, 0]]
        sage: sorted(I.orbit([2,0,2,0]))
        [[0, 2, 0, 2], [2, 0, 2, 0]]
        sage: I.orbit([1,1,1,1])
        {[1, 1, 1, 1]}

  `IntegerVectorsModPermutationGroup_All.``permutation_group`()

    Returns the permutation group given to define `self`.

    EXAMPLES:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: I.permutation_group()
        Permutation Group with generators [(1,2,3,4)]

  `IntegerVectorsModPermutationGroup_All.``retract`(elt)

    Return the canonical representative of the orbit of the integer `elt` under the action of the permutation group defining `self`.

    If the element `elt` is already maximal in its orbit for the lexicographic order, `elt` is thus the good representative for its orbit.

    EXAMPLES:

        sage: [0,0,0,0] in IntegerVectors(length=4)
        True
        sage: [1,0,0,0] in IntegerVectors(length=4)
        True
        sage: [0,1,0,0] in IntegerVectors(length=4)
        True
        sage: [1,0,1,0] in IntegerVectors(length=4)
        True
        sage: [0,1,0,1] in IntegerVectors(length=4)
        True
        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: S.retract([0,0,0,0])
        [0, 0, 0, 0]
        sage: S.retract([1,0,0,0])
        [1, 0, 0, 0]
        sage: S.retract([0,1,0,0])
        [1, 0, 0, 0]
        sage: S.retract([1,0,1,0])
        [1, 0, 1, 0]
        sage: S.retract([0,1,0,1])
        [1, 0, 1, 0]

  `IntegerVectorsModPermutationGroup_All.``roots`()

    Returns the root of generation of `self`. This method is required to build the tree structure of `self` which inherits from the class `SearchForest`.

    EXAMPLES:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: I.roots()
        [[0, 0, 0, 0]]

  `IntegerVectorsModPermutationGroup_All.``subset`(sum=None, max_part=None)

    Returns the subset of `self` containing integer vectors whose entries sum to `sum`.

    EXAMPLES:

        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: S.subset(4)
        Integer vectors of length 4 and of sum 4 enumerated up to
        the action of Permutation Group with generators
        [(1,2,3,4)]

class `sage.combinat.integer_vectors_mod_permgroup.``IntegerVectorsModPermutationGroup_with_constraints`(G, d, max_part, sgs=None)

  Bases: `sage.structure.unique_representation.UniqueRepresentation`, `sage.combinat.backtrack.SearchForest`

  This class models finite enumerated sets of integer vectors with constraint enumerated up to the action of a permutation group. Integer vectors are enumerated modulo the action of the permutation group. To implement that, we keep a single integer vector by orbit under the action of the permutation group. Elements chosen are vectors maximal in their orbit for the lexicographic order.

  For more information see `IntegerVectorsModPermutationGroup`.

  EXAMPLES:

      sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), max_part=1)
      sage: I.list()
      [[0, 0, 0, 0], [1, 0, 0, 0], [1, 1, 0, 0], [1, 0, 1, 0], [1, 1, 1, 0], [1, 1, 1, 1]]
      sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), sum=6, max_part=4)
      sage: I.list()
      [[4, 2, 0, 0], [4, 1, 1, 0], [4, 1, 0, 1], [4, 0, 2, 0], [4, 0, 1, 1],
       [4, 0, 0, 2], [3, 3, 0, 0], [3, 2, 1, 0], [3, 2, 0, 1], [3, 1, 2, 0],
       [3, 1, 1, 1], [3, 1, 0, 2], [3, 0, 3, 0], [3, 0, 2, 1], [3, 0, 1, 2],
       [2, 2, 2, 0], [2, 2, 1, 1], [2, 1, 2, 1]]

  Here is the enumeration of unlabeled graphs over 5 vertices:

      sage: G = IntegerVectorsModPermutationGroup(TransitiveGroup(10,12), max_part=1) # optional
      sage: G.cardinality() # optional
      34

  TESTS:

      sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]),4)
      sage: TestSuite(I).run()

  class `Element`

    Bases: `sage.structure.list_clone.ClonableIntArray`

    Element class for the set of integer vectors with constraints enumerated modulo the action of a permutation group. These vectors are clonable lists of integers which must check conditions comming form the parent as in the method `check()`.

    TESTS:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), 4)
        sage: v = I.element_class(I, [3,1,0,0]); v
        [3, 1, 0, 0]
        sage: TestSuite(v).run()
        sage: v = I.element_class(I, [3,2,0,0])
        Traceback (most recent call last):
        ...
        AssertionError: [3, 2, 0, 0] should be a integer vector of sum 4

    `check`()

      Checks that `self` meets the constraints of being an element of `self.parent()`.

      EXAMPLES:

          sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), 4)
          sage: v = I.an_element()
          sage: v.check()
          sage: w = I([0,4,0,0], check=False); w
          [0, 4, 0, 0]
          sage: w.check()
          Traceback (most recent call last):
          ...
          AssertionError

  `IntegerVectorsModPermutationGroup_with_constraints.``ambient`()

    Return the ambient space from which `self` is a quotient.

    EXAMPLES:

        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), 6); S.ambient()
        Integer vectors that sum to 6
        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), 6, max_part=12); S.ambient()
        Integer vectors that sum to 6 with constraints: max_part=12

    Todo

    Integer vectors should accept `max_part` as a single argument, and the following should change:

        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), max_part=12); S.ambient()
        Integer vectors

  `IntegerVectorsModPermutationGroup_with_constraints.``an_element`()

    Returns an element of `self` or raises an EmptySetError when `self` is empty.

    EXAMPLES:

        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), sum=0, max_part=1); S.an_element()
        [0, 0, 0, 0]
        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), sum=1, max_part=1); S.an_element()
        [1, 0, 0, 0]
        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), sum=2, max_part=1); S.an_element()
        [1, 1, 0, 0]
        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), sum=3, max_part=1); S.an_element()
        [1, 1, 1, 0]
        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), sum=4, max_part=1); S.an_element()
        [1, 1, 1, 1]
        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), sum=5, max_part=1); S.an_element()
        Traceback (most recent call last):
        ...
        EmptySetError

  `IntegerVectorsModPermutationGroup_with_constraints.``children`(x)

    Returns the list of children of the element `x`. This method is required to build the tree structure of `self` which inherits from the class `SearchForest`.

    EXAMPLES:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: I.children(I([2,1,0,0], check=False))
        [[2, 2, 0, 0], [2, 1, 1, 0], [2, 1, 0, 1]]

  `IntegerVectorsModPermutationGroup_with_constraints.``is_canonical`(v, check=True)

    Returns `True` if the integer list `v` is maximal in its orbit under the action of the permutation group given to define `self`. Such integer vectors are said to be canonical. A vector \(v\) is canonical if and only if

    \[v = \max_{\text{lex order}} \{g \cdot v | g \in G \}\]

    EXAMPLES:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), max_part=3)
        sage: I.is_canonical([3,0,0,0])
        True
        sage: I.is_canonical([1,0,2,0])
        False
        sage: I.is_canonical([2,0,1,0])
        True

  `IntegerVectorsModPermutationGroup_with_constraints.``lift`(elt)

    Lift the element `elt` inside the ambient space from which `self` is a quotient.

    EXAMPLES:

        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), max_part=1)
        sage: v = S.lift([1,0,1,0]); v
        [1, 0, 1, 0]
        sage: v in IntegerVectors(max_part=1)
        True
        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), sum=6)
        sage: v = S.lift(S.list()[5]); v
        [4, 1, 1, 0]
        sage: v in IntegerVectors(n=6)
        True

  `IntegerVectorsModPermutationGroup_with_constraints.``orbit`(v)

    Returns the orbit of the vector `v` under the action of the permutation group defining `self`. The result is a set.

    INPUT:

    - `v` - an element of `self` or any list of length the degree of the permutation group.

    EXAMPLES:

    We convert the result in a list in increasing lexicographic order, to get a reproducible doctest:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]),4)
        sage: I.orbit([1,1,1,1])
        {[1, 1, 1, 1]}
        sage: sorted(I.orbit([3,0,0,1]))
        [[0, 0, 1, 3], [0, 1, 3, 0], [1, 3, 0, 0], [3, 0, 0, 1]]

  `IntegerVectorsModPermutationGroup_with_constraints.``permutation_group`()

    Returns the permutation group given to define `self`.

    EXAMPLES:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3)]]), 5)
        sage: I.permutation_group()
        Permutation Group with generators [(1,2,3)]

  `IntegerVectorsModPermutationGroup_with_constraints.``retract`(elt)

    Return the canonical representative of the orbit of the integer `elt` under the action of the permutation group defining `self`.

    If the element `elt` is already maximal in its orbits for the lexicographic order, `elt` is thus the good representative for its orbit.

    EXAMPLES:

        sage: S = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]), sum=2, max_part=1)
        sage: S.retract([1,1,0,0])
        [1, 1, 0, 0]
        sage: S.retract([1,0,1,0])
        [1, 0, 1, 0]
        sage: S.retract([1,0,0,1])
        [1, 1, 0, 0]
        sage: S.retract([0,1,1,0])
        [1, 1, 0, 0]
        sage: S.retract([0,1,0,1])
        [1, 0, 1, 0]
        sage: S.retract([0,0,1,1])
        [1, 1, 0, 0]

  `IntegerVectorsModPermutationGroup_with_constraints.``roots`()

    Returns the root of generation of `self`.This method is required to build the tree structure of `self` which inherits from the class `SearchForest`.

    EXAMPLES:

        sage: I = IntegerVectorsModPermutationGroup(PermutationGroup([[(1,2,3,4)]]))
        sage: I.roots()
        [[0, 0, 0, 0]]

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