# Utilities¶

sympy.combinatorics.util._base_ordering(base, degree)[source]

Order $$\{0, 1, ..., n-1\}$$ so that base points come first and in order.

Parameters : base - the base degree - the degree of the associated permutation group A list base_ordering such that base_ordering[point] is the number of point in the ordering.

Notes

This is used in backtrack searches, when we define a relation $$<<$$ on the underlying set for a permutation group of degree $$n$$, $$\{0, 1, ..., n-1\}$$, so that if $$(b_1, b_2, ..., b_k)$$ is a base we have $$b_i << b_j$$ whenever $$i<j$$ and $$b_i << a$$ for all $$i\in\{1,2, ..., k\}$$ and $$a$$ is not in the base. The idea is developed and applied to backtracking algorithms in [1], pp.108-132. The points that are not in the base are taken in increasing order.

References

[1] Holt, D., Eick, B., O’Brien, E. “Handbook of computational group theory”

Examples

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.util import _base_ordering
>>> S = SymmetricGroup(4)
>>> S.schreier_sims()
>>> _base_ordering(S.base, S.degree)
[0, 1, 2, 3]

sympy.combinatorics.util._check_cycles_alt_sym(perm)[source]

Checks for cycles of prime length p with n/2 < p < n-2.

Here $$n$$ is the degree of the permutation. This is a helper function for the function is_alt_sym from sympy.combinatorics.perm_groups.

Examples

>>> from sympy.combinatorics.util import _check_cycles_alt_sym
>>> from sympy.combinatorics.permutations import Permutation
>>> a = Permutation([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10], [11, 12]])
>>> _check_cycles_alt_sym(a)
False
>>> b = Permutation([[0, 1, 2, 3, 4, 5, 6], [7, 8, 9, 10]])
>>> _check_cycles_alt_sym(b)
True

sympy.combinatorics.util._distribute_gens_by_base(base, gens)[source]

Distribute the group elements gens by membership in basic stabilizers.

Notice that for a base $$(b_1, b_2, ..., b_k)$$, the basic stabilizers are defined as $$G^{(i)} = G_{b_1, ..., b_{i-1}}$$ for $$i \in\{1, 2, ..., k\}$$.

Parameters : base - a sequence of points in {0, 1, ..., n-1} gens - a list of elements of a permutation group of degree n. List of length $$k$$, where $$k$$ is the length of base. The $$i$$-th entry contains those elements in gens which fix the first $$i$$ elements of base (so that the $$0$$-th entry is equal to gens itself). If no element fixes the first $$i$$ elements of base, the $$i$$-th element is set to a list containing the identity element.

Examples

>>> from sympy.combinatorics import Permutation
>>> Permutation.print_cyclic = True
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> from sympy.combinatorics.util import _distribute_gens_by_base
>>> D = DihedralGroup(3)
>>> D.schreier_sims()
>>> D.strong_gens
[(0 1 2), (0 2), (1 2)]
>>> D.base
[0, 1]
>>> _distribute_gens_by_base(D.base, D.strong_gens)
[[(0 1 2), (0 2), (1 2)],
[(1 2)]]

sympy.combinatorics.util._handle_precomputed_bsgs(base, strong_gens, transversals=None, basic_orbits=None, strong_gens_distr=None)[source]

Calculate BSGS-related structures from those present.

The base and strong generating set must be provided; if any of the transversals, basic orbits or distributed strong generators are not provided, they will be calculated from the base and strong generating set.

Parameters : base - the base strong_gens - the strong generators transversals - basic transversals basic_orbits - basic orbits strong_gens_distr - strong generators distributed by membership in basic stabilizers (transversals, basic_orbits, strong_gens_distr) where transversals are the basic transversals, basic_orbits are the basic orbits, and strong_gens_distr are the strong generators distributed by membership in basic stabilizers.

_orbits_transversals_from_bsgs, distribute_gens_by_base

Examples

>>> from sympy.combinatorics import Permutation
>>> Permutation.print_cyclic = True
>>> from sympy.combinatorics.named_groups import DihedralGroup
>>> from sympy.combinatorics.util import _handle_precomputed_bsgs
>>> D = DihedralGroup(3)
>>> D.schreier_sims()
>>> _handle_precomputed_bsgs(D.base, D.strong_gens,
... basic_orbits=D.basic_orbits)
([{0: (2), 1: (0 1 2), 2: (0 2)}, {1: (2), 2: (1 2)}], [[0, 1, 2], [1, 2]], [[(0 1 2), (0 2), (1 2)], [(1 2)]])

sympy.combinatorics.util._orbits_transversals_from_bsgs(base, strong_gens_distr, transversals_only=False)[source]

Compute basic orbits and transversals from a base and strong generating set.

The generators are provided as distributed across the basic stabilizers. If the optional argument transversals_only is set to True, only the transversals are returned.

Parameters : base - the base strong_gens_distr - strong generators distributed by membership in basic stabilizers transversals_only - a flag switching between returning only the transversals/ both orbits and transversals

Examples

>>> from sympy.combinatorics import Permutation
>>> Permutation.print_cyclic = True
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.util import _orbits_transversals_from_bsgs
>>> from sympy.combinatorics.util import (_orbits_transversals_from_bsgs,
... _distribute_gens_by_base)
>>> S = SymmetricGroup(3)
>>> S.schreier_sims()
>>> strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens)
>>> _orbits_transversals_from_bsgs(S.base, strong_gens_distr)
([[0, 1, 2], [1, 2]], [{0: (2), 1: (0 1 2), 2: (0 2 1)}, {1: (2), 2: (1 2)}])

sympy.combinatorics.util._remove_gens(base, strong_gens, basic_orbits=None, strong_gens_distr=None)[source]

Remove redundant generators from a strong generating set.

Parameters : base - a base strong_gens - a strong generating set relative to base basic_orbits - basic orbits strong_gens_distr - strong generators distributed by membership in basic stabilizers A strong generating set with respect to base which is a subset of strong_gens.

Notes

This procedure is outlined in [1],p.95.

References

[1] Holt, D., Eick, B., O’Brien, E. “Handbook of computational group theory”

Examples

>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.perm_groups import PermutationGroup
>>> from sympy.combinatorics.util import _remove_gens
>>> from sympy.combinatorics.testutil import _verify_bsgs
>>> S = SymmetricGroup(15)
>>> base, strong_gens = S.schreier_sims_incremental()
>>> new_gens = _remove_gens(base, strong_gens)
>>> len(new_gens)
14
>>> _verify_bsgs(S, base, new_gens)
True

sympy.combinatorics.util._strip(g, base, orbits, transversals)[source]

Attempt to decompose a permutation using a (possibly partial) BSGS structure.

This is done by treating the sequence base as an actual base, and the orbits orbits and transversals transversals as basic orbits and transversals relative to it.

This process is called “sifting”. A sift is unsuccessful when a certain orbit element is not found or when after the sift the decomposition doesn’t end with the identity element.

The argument transversals is a list of dictionaries that provides transversal elements for the orbits orbits.

Parameters : g - permutation to be decomposed base - sequence of points orbits - a list in which the i-th entry is an orbit of base[i] under some subgroup of the pointwise stabilizer of  base[0], base[1], ..., base[i - 1]. The groups themselves are implicit in this function since the only information we need is encoded in the orbits and transversals transversals - a list of orbit transversals associated with the orbits orbits.

Notes

The algorithm is described in [1],pp.89-90. The reason for returning both the current state of the element being decomposed and the level at which the sifting ends is that they provide important information for the randomized version of the Schreier-Sims algorithm.

References

[1] Holt, D., Eick, B., O’Brien, E. “Handbook of computational group theory”

Examples

>>> from sympy.combinatorics import Permutation
>>> Permutation.print_cyclic = True
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.permutations import Permutation
>>> from sympy.combinatorics.util import _strip
>>> S = SymmetricGroup(5)
>>> S.schreier_sims()
>>> g = Permutation([0, 2, 3, 1, 4])
>>> _strip(g, S.base, S.basic_orbits, S.basic_transversals)
((4), 5)

sympy.combinatorics.util._strong_gens_from_distr(strong_gens_distr)[source]

Retrieve strong generating set from generators of basic stabilizers.

This is just the union of the generators of the first and second basic stabilizers.

Parameters : strong_gens_distr - strong generators distributed by membership in basic stabilizers

Examples

>>> from sympy.combinatorics import Permutation
>>> Permutation.print_cyclic = True
>>> from sympy.combinatorics.named_groups import SymmetricGroup
>>> from sympy.combinatorics.util import (_strong_gens_from_distr,
... _distribute_gens_by_base)
>>> S = SymmetricGroup(3)
>>> S.schreier_sims()
>>> S.strong_gens
[(0 1 2), (2)(0 1), (1 2)]
>>> strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens)
>>> _strong_gens_from_distr(strong_gens_distr)
[(0 1 2), (2)(0 1), (1 2)]


Named Groups

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