Utilities#

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

Order \(\{0, 1, \dots, n-1\}\) 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.

Examples

>>> from sympy.combinatorics 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]

Notes

This is used in backtrack searches, when we define a relation \(\ll\) on the underlying set for a permutation group of degree \(n\), \(\{0, 1, \dots, n-1\}\), so that if \((b_1, b_2, \dots, b_k)\) is a base we have \(b_i \ll b_j\) whenever \(i<j\) and \(b_i \ll a\) for all \(i\in\{1,2, \dots, k\}\) 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

R52: Holt, D., Eick, B., O’Brien, E. “Handbook of computational group theory”

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

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

Explanation

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 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.

Parameters

``base`` : a sequence of points in \(\{0, 1, \dots, n-1\}\)

``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.

Explanation

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

Examples

>>> 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.

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.

Explanation

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.

Examples

>>> 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.

Parameters

``base`` - The base.

``strong_gens_distr`` - Strong generators distributed by membership in basic

stabilizers.

``transversals_only`` - bool

A flag switching between returning only the transversals and 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.

Explanation

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.

Examples

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

Examples

>>> from sympy.combinatorics import SymmetricGroup
>>> 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

Notes

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

References

R53: Holt, D., Eick, B., O’Brien, E. “Handbook of computational group theory”

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

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

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``.

Explanation

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.

Examples

>>> from sympy.combinatorics import Permutation, SymmetricGroup
>>> 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)

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

R54: Holt, D., Eick, B., O’Brien, E.”Handbook of computational group theory”

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 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)]

See also

_distribute_gens_by_base