# Tools for enumeration modulo the action of a permutation group¶

Tools for enumeration modulo the action of a permutation group

sage.combinat.enumeration_mod_permgroup.all_children(v, max_part)

Returns all the children of an integer vector (ClonableIntArray) v in the tree of enumeration by lexicographic order. The children of an integer vector v whose entries have the sum $$n$$ are all integer vectors of sum $$n+1$$ which follow v in the lexicographic order.

That means this function adds $$1$$ on the last non zero entries and the following ones. For an integer vector $$v$$ such that

$v = [ \dots, a , 0, 0 ] \text{ with } a \neq 0,$

then, the list of children is

$[ [ \dots, a+1 , 0, 0 ] , [ \dots, a , 1, 0 ], [ \dots, a , 0, 1 ] ].$

EXAMPLES:

sage: from sage.combinat.enumeration_mod_permgroup import all_children
sage: from sage.structure.list_clone_demo import IncreasingIntArrays
sage: v = IncreasingIntArrays()([1,2,3,4])
sage: all_children(v, -1)
[[1, 2, 3, 5]]

sage.combinat.enumeration_mod_permgroup.canonical_children(sgs, v, max_part)

Returns the canonical children of the integer vector v. This function computes all children of the integer vector v via the function all_children() and returns from this list only these which are canonicals identified via the function is_canonical().

EXAMPLES:

sage: from sage.combinat.enumeration_mod_permgroup import canonical_children
sage: G = PermutationGroup([[(1,2,3,4)]])
sage: sgs = G.strong_generating_system()
sage: from sage.structure.list_clone_demo import IncreasingIntArrays
sage: IA = IncreasingIntArrays()
sage: canonical_children(sgs, IA([1,2,3,5]), -1)
[]

sage.combinat.enumeration_mod_permgroup.canonical_representative_of_orbit_of(sgs, v)

Returns the maximal vector for the lexicographic order living in the orbit of $$v$$ under the action of the permutation group whose strong generating system is sgs. The maximal vector is also called “canonical”. Hence, this method returns the canonical vector inside the orbit of $$v$$. If $$v$$ is already canonical, the method returns $$v$$.

Let $$G$$ to be the permutation group which admits sgs as a strong generating system. An integer vector $$v$$ is said to be canonical under the action of $$G$$ if and only if:

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

EXAMPLES:

sage: from sage.combinat.enumeration_mod_permgroup import canonical_representative_of_orbit_of
sage: G = PermutationGroup([[(1,2,3,4)]])
sage: sgs = G.strong_generating_system()
sage: from sage.structure.list_clone_demo import IncreasingIntArrays
sage: IA = IncreasingIntArrays()
sage: canonical_representative_of_orbit_of(sgs, IA([1,2,3,5]))
[5, 1, 2, 3]

sage.combinat.enumeration_mod_permgroup.is_canonical(sgs, v)

Returns True if the integer vector $$v$$ is maximal with respect to the lexicographic order in its orbit under the action of the permutation group whose strong generating system is sgs. Such vectors are said to be canonical.

Let $$G$$ to be the permutation group which admit sgs as a strong generating system. An integer vector $$v$$ is said to be canonical under the action of $$G$$ if and only if:

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

EXAMPLES:

sage: from sage.combinat.enumeration_mod_permgroup import is_canonical
sage: G = PermutationGroup([[(1,2,3,4)]])
sage: sgs = G.strong_generating_system()
sage: from sage.structure.list_clone_demo import IncreasingIntArrays
sage: IA = IncreasingIntArrays()
sage: is_canonical(sgs, IA([1,2,3,6]))
False

sage.combinat.enumeration_mod_permgroup.lex_cmp(v1, v2)

Lexicographic comparison of ClonableIntArray.

INPUT:

Two instances $$v_1, v_2$$ of ClonableIntArray

OUPUT:

-1,0,1, depending on whether $$v_1$$ is lexicographically smaller, equal, or greater than $$v_2$$.

EXAMPLES:

sage: I =  IntegerVectorsModPermutationGroup(SymmetricGroup(5),5)
sage: I =  IntegerVectorsModPermutationGroup(SymmetricGroup(5))
sage: J =  IntegerVectorsModPermutationGroup(SymmetricGroup(6))
sage: v1 = I([2,3,1,2,3], check=False)
sage: v2 = I([2,3,2,3,2], check=False)
sage: v3 = J([2,3,1,2,3,1], check=False)
sage: from sage.combinat.enumeration_mod_permgroup import lex_cmp
sage: lex_cmp(v1, v1)
0
sage: lex_cmp(v1, v2)
-1
sage: lex_cmp(v2, v1)
1
sage: lex_cmp(v1, v3)
-1
sage: lex_cmp(v3, v1)
1

sage.combinat.enumeration_mod_permgroup.lex_cmp_partial(v1, v2, step)

Partial comparison of the two lists according the lexicographic order. It compares the step-th first entries.

EXAMPLES:

sage: from sage.combinat.enumeration_mod_permgroup import lex_cmp_partial
sage: from sage.structure.list_clone_demo import IncreasingIntArrays
sage: IA = IncreasingIntArrays()
sage: lex_cmp_partial(IA([0,1,2,3]),IA([0,1,2,4]),3)
0
sage: lex_cmp_partial(IA([0,1,2,3]),IA([0,1,2,4]),4)
-1

sage.combinat.enumeration_mod_permgroup.orbit(sgs, v)

Returns the orbit of the integer vector v under the action of the permutation group whose strong generating system is sgs.

NOTE:

The returned orbit is a set. In the doctests, we convert it into a sorted list.

EXAMPLES:

sage: from sage.combinat.enumeration_mod_permgroup import orbit, lex_cmp
sage: G = PermutationGroup([[(1,2,3,4)]])
sage: sgs = G.strong_generating_system()
sage: from sage.structure.list_clone_demo import IncreasingIntArrays
sage: IA = IncreasingIntArrays()
sage: sorted(orbit(sgs, IA([1,2,3,4])), cmp=lex_cmp)
[[1, 2, 3, 4], [2, 3, 4, 1], [3, 4, 1, 2], [4, 1, 2, 3]]


#### Previous topic

Integer vectors modulo the action of a permutation group

Hall Polynomials