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

Returns

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.

See also

sympy.combinatorics.perm_groups.PermutationGroup.is_alt_sym

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`.

Returns

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.

See also

_strong_gens_from_distr, _orbits_transversals_from_bsgs, _handle_precomputed_bsgs

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

Returns

(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.

See also

_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, slp=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

``slp`` - if ``True``, return a list of dictionaries containing the

generator presentations of the elements of the transversals, i.e. the list of indices of generators from \(strong_gens_distr[i]\) such that their product is the relevant transversal element

See also

_distribute_gens_by_base, _handle_precomputed_bsgs

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

Returns

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``.

See also

sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims, sympy.combinatorics.perm_groups.PermutationGroup.schreier_sims_random

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

See also

_distribute_gens_by_base

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)]