Utilities¶

diofant.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.

Examples

>>> 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 $$<<$$ 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 , pp.108-132. The points that are not in the base are taken in increasing order.

References

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

diofant.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 diofant.combinatorics.perm_groups.

Examples

>>> 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

diofant.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.

Examples

>>> Permutation.print_cyclic = True
>>> D = DihedralGroup(3)
>>> D.schreier_sims()
>>> D.strong_gens
[Permutation(0, 1, 2), Permutation(0, 2), Permutation(1, 2)]
>>> D.base
[0, 1]
>>> _distribute_gens_by_base(D.base, D.strong_gens)
[[Permutation(0, 1, 2), Permutation(0, 2), Permutation(1, 2)],
[Permutation(1, 2)]]

diofant.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.

Examples

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

diofant.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

>>> Permutation.print_cyclic = True
>>> 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: Permutation(2), 1: Permutation(0, 1, 2), 2: Permutation(0, 2, 1)},
{1: Permutation(2), 2: Permutation(1, 2)}])

diofant.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 diofant.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 ,p.95.

References

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

diofant.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, base, …, 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.

Examples

>>> Permutation.print_cyclic = True
>>> S = SymmetricGroup(5)
>>> S.schreier_sims()
>>> g = Permutation([0, 2, 3, 1, 4])
>>> _strip(g, S.base, S.basic_orbits, S.basic_transversals)
(Permutation(4), 5)


Notes

The algorithm is described in ,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

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

diofant.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

>>> Permutation.print_cyclic = True
>>> S = SymmetricGroup(3)
>>> S.schreier_sims()
>>> S.strong_gens
[Permutation(0, 1, 2), Permutation(2)(0, 1), Permutation(1, 2)]
>>> strong_gens_distr = _distribute_gens_by_base(S.base, S.strong_gens)
>>> _strong_gens_from_distr(strong_gens_distr)
[Permutation(0, 1, 2), Permutation(2)(0, 1), Permutation(1, 2)]