Set Species

sage.combinat.species.set_species.SetSpecies(*args, **kwds)

Returns the species of sets.

EXAMPLES:

sage: E = species.SetSpecies()
sage: E.structures([1,2,3]).list()
[{1, 2, 3}]
sage: E.isotype_generating_series().coefficients(4)
[1, 1, 1, 1]
class sage.combinat.species.set_species.SetSpeciesStructure(parent, labels, list)
__repr__()

EXAMPLES:

sage: S = species.SetSpecies()
sage: a = S.structures(["a","b","c"]).random_element(); a
{'a', 'b', 'c'}
automorphism_group()

Returns the group of permutations whose action on this set leave it fixed. For the species of sets, there is only one isomorphism class, so every permutation is in its automorphism group.

EXAMPLES:

sage: F = species.SetSpecies()
sage: a = F.structures(["a", "b", "c"]).random_element(); a
{'a', 'b', 'c'}
sage: a.automorphism_group()
Symmetric group of order 3! as a permutation group
canonical_label()

EXAMPLES:

sage: S = species.SetSpecies()
sage: a = S.structures(["a","b","c"]).random_element(); a
{'a', 'b', 'c'}
sage: a.canonical_label()
{'a', 'b', 'c'}
transport(perm)

Returns the transport of this set along the permutation perm.

EXAMPLES:

sage: F = species.SetSpecies()
sage: a = F.structures(["a", "b", "c"]).random_element(); a
{'a', 'b', 'c'}
sage: p = PermutationGroupElement((1,2))
sage: a.transport(p)
{'a', 'b', 'c'}
class sage.combinat.species.set_species.SetSpecies_class(min=None, max=None, weight=None)
__init__(min=None, max=None, weight=None)

EXAMPLES:

sage: S = species.SetSpecies()
sage: c = S.generating_series().coefficients(3)
sage: S._check()
True
sage: S == loads(dumps(S))
True
static _cached_constructor(*args, **kwds)

Returns the species of sets.

EXAMPLES:

sage: E = species.SetSpecies()
sage: E.structures([1,2,3]).list()
[{1, 2, 3}]
sage: E.isotype_generating_series().coefficients(4)
[1, 1, 1, 1]
_cis(series_ring, base_ring)

The cycle index series for the species of sets is given by

System Message: WARNING/2 (exp\( \sum_{n=1}{\infty} = \frac{x_n}{n} \))

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.

EXAMPLES:

sage: S = species.SetSpecies()
sage: g = S.cycle_index_series()
sage: g.coefficients(5)
[p[],
 p[1],
 1/2*p[1, 1] + 1/2*p[2],
 1/6*p[1, 1, 1] + 1/2*p[2, 1] + 1/3*p[3],
 1/24*p[1, 1, 1, 1] + 1/4*p[2, 1, 1] + 1/8*p[2, 2] + 1/3*p[3, 1] + 1/4*p[4]]
_cis_gen(base_ring)

EXAMPLES:

sage: S = species.SetSpecies()
sage: g = S._cis_gen(QQ)
sage: [g.next() for i in range(5)]
[0, p[1], 1/2*p[2], 1/3*p[3], 1/4*p[4]]
_default_structure_class
alias of SetSpeciesStructure
_gs_iterator(base_ring)

The generating series for the species of sets is given by e^x.

EXAMPLES:

sage: S = species.SetSpecies()
sage: g = S.generating_series()
sage: g.coefficients(10)
[1, 1, 1/2, 1/6, 1/24, 1/120, 1/720, 1/5040, 1/40320, 1/362880]
sage: [g.count(i) for i in range(10)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
_isotypes(structure_class, labels)

EXAMPLES:

sage: S = species.SetSpecies()
sage: S.structures([1,2,3]).list()
[{1, 2, 3}]
_itgs_list(base_ring)

The isomorphism type generating series for the species of sets is \frac{1}{1-x}.

EXAMPLES:

sage: S = species.SetSpecies()
sage: g = S.isotype_generating_series()
sage: g.coefficients(10)
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
sage: [g.count(i) for i in range(10)]
[1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
_structures(structure_class, labels)

EXAMPLES:

sage: S = species.SetSpecies()
sage: S.structures([1,2,3]).list()
[{1, 2, 3}]

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