This file provides the cycle index series for the virtual species ,
the ‘combinatorial logarithm’, defined to be the compositional inverse of
the species
of nonempty sets:
AUTHORS:
TESTS:
sage: from sage.combinat.species.combinatorial_logarithm import CombinatorialLogarithmSeries
sage: CombinatorialLogarithmSeries().coefficients(5)
[0, p[1], -1/2*p[1, 1] - 1/2*p[2], 1/3*p[1, 1, 1] - 1/3*p[3], -1/4*p[1, 1, 1, 1] + 1/4*p[2, 2]]
sage: Eplus = sage.combinat.species.set_species.SetSpecies(min=1).cycle_index_series()
sage: CombinatorialLogarithmSeries().compose(Eplus).coefficients(4)
[0, p[1], 0, 0]
Return the cycle index series of the virtual species , the compositional inverse
of the species
of nonempty sets.
The notion of virtual species is treated thoroughly in [BLL]. The specific algorithm used
here to compute the cycle index of is found in [Labelle].
EXAMPLES:
The virtual species is ‘properly virtual’, in the sense that its cycle index
has negative coefficients:
sage: from sage.combinat.species.combinatorial_logarithm import CombinatorialLogarithmSeries
sage: CombinatorialLogarithmSeries().coefficients(4)
[0, p[1], -1/2*p[1, 1] - 1/2*p[2], 1/3*p[1, 1, 1] - 1/3*p[3]]
Its defining property is that (that is, that
composition with
in both directions yields the multiplicative identity
):
sage: Eplus = sage.combinat.species.set_species.SetSpecies(min=1).cycle_index_series()
sage: CombinatorialLogarithmSeries().compose(Eplus).coefficients(4)
[0, p[1], 0, 0]
REFERENCES:
[BLL] |
|
[Labelle] |
|