Empty Species#
- class sage.combinat.species.empty_species.EmptySpecies(min=None, max=None, weight=None)#
Bases:
sage.combinat.species.species.GenericCombinatorialSpecies,sage.structure.unique_representation.UniqueRepresentationReturns the empty species. This species has no structure at all. It is the zero of the semi-ring of species.
EXAMPLES:
sage: X = species.EmptySpecies(); X Empty species sage: X.structures([]).list() [] sage: X.structures([1]).list() [] sage: X.structures([1,2]).list() [] sage: X.generating_series()[0:4] [0, 0, 0, 0] sage: X.isotype_generating_series()[0:4] [0, 0, 0, 0] sage: X.cycle_index_series()[0:4] [0, 0, 0, 0]
The empty species is the zero of the semi-ring of species. The following tests that it is neutral with respect to addition:
sage: Empt = species.EmptySpecies() sage: S = species.CharacteristicSpecies(2) sage: X = S + Empt sage: X == S # TODO: Not Implemented True sage: (X.generating_series()[0:4] == ....: S.generating_series()[0:4]) True sage: (X.isotype_generating_series()[0:4] == ....: S.isotype_generating_series()[0:4]) True sage: (X.cycle_index_series()[0:4] == ....: S.cycle_index_series()[0:4]) True
The following tests that it is the zero element with respect to multiplication:
sage: Y = Empt*S sage: Y == Empt # TODO: Not Implemented True sage: Y.generating_series()[0:4] [0, 0, 0, 0] sage: Y.isotype_generating_series()[0:4] [0, 0, 0, 0] sage: Y.cycle_index_series()[0:4] [0, 0, 0, 0]
- sage.combinat.species.empty_species.EmptySpecies_class#