Empty Species

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

Bases: sage.combinat.species.species.GenericCombinatorialSpecies

Returns the empty species. This species has no structure at all. It is the zeros or 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().coefficients(4)
[0, 0, 0, 0]
sage: X.isotype_generating_series().coefficients(4)
[0, 0, 0, 0]
sage: X.cycle_index_series().coefficients(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().coefficients(4) ==
...    S.generating_series().coefficients(4))
True
sage: (X.isotype_generating_series().coefficients(4) ==
...    S.isotype_generating_series().coefficients(4))
True
sage: (X.cycle_index_series().coefficients(4) ==
...    S.cycle_index_series().coefficients(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().coefficients(4)
[0, 0, 0, 0]
sage: Y.isotype_generating_series().coefficients(4)
[0, 0, 0, 0]
sage: Y.cycle_index_series().coefficients(4)
[0, 0, 0, 0]

TESTS:

sage: Empt  = species.EmptySpecies()
sage: Empt2 = species.EmptySpecies()
sage: Empt is Empt2
True