Empty Species

class sage.combinat.species.empty_species.EmptySpecies(min=None, max=None, weight=None)

Bases: GenericCombinatorialSpecies, UniqueRepresentation

Returns 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

alias of EmptySpecies