Empty Species#
- class sage.combinat.species.empty_species.EmptySpecies(min=None, max=None, weight=None)[source]#
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] # needs sage.modules [0, 0, 0, 0]
>>> from sage.all import * >>> X = species.EmptySpecies(); X Empty species >>> X.structures([]).list() [] >>> X.structures([Integer(1)]).list() [] >>> X.structures([Integer(1),Integer(2)]).list() [] >>> X.generating_series()[Integer(0):Integer(4)] [0, 0, 0, 0] >>> X.isotype_generating_series()[Integer(0):Integer(4)] [0, 0, 0, 0] >>> X.cycle_index_series()[Integer(0):Integer(4)] # needs sage.modules [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] == # needs sage.modules ....: S.cycle_index_series()[0:4]) True
>>> from sage.all import * >>> Empt = species.EmptySpecies() >>> S = species.CharacteristicSpecies(Integer(2)) >>> X = S + Empt >>> X == S # TODO: Not Implemented True >>> (X.generating_series()[Integer(0):Integer(4)] == ... S.generating_series()[Integer(0):Integer(4)]) True >>> (X.isotype_generating_series()[Integer(0):Integer(4)] == ... S.isotype_generating_series()[Integer(0):Integer(4)]) True >>> (X.cycle_index_series()[Integer(0):Integer(4)] == # needs sage.modules ... S.cycle_index_series()[Integer(0):Integer(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] # needs sage.modules [0, 0, 0, 0]
>>> from sage.all import * >>> Y = Empt*S >>> Y == Empt # TODO: Not Implemented True >>> Y.generating_series()[Integer(0):Integer(4)] [0, 0, 0, 0] >>> Y.isotype_generating_series()[Integer(0):Integer(4)] [0, 0, 0, 0] >>> Y.cycle_index_series()[Integer(0):Integer(4)] # needs sage.modules [0, 0, 0, 0]
- sage.combinat.species.empty_species.EmptySpecies_class[source]#
alias of
EmptySpecies