Combinatorial Logarithm¶
This file provides the cycle index series for the virtual species \(\Omega\), the ‘combinatorial logarithm’, defined to be the compositional inverse of the species \(E^{+}\) of nonempty sets:
Warning
This module is now deprecated. Please use
sage.combinat.species.generating_series.CycleIndexSeriesRing.exponential()
instead of CombinatorialLogarithmSeries()
.
AUTHORS:
- Andrew Gainer-Dewar (2013): initial version
-
sage.combinat.species.combinatorial_logarithm.
CombinatorialLogarithmSeries
(R=Rational Field)¶ Return the cycle index series of the virtual species \(\Omega\), the compositional inverse of the species \(E^{+}\) of nonempty sets.
The notion of virtual species is treated thoroughly in [BLL]. The specific algorithm used here to compute the cycle index of \(\Omega\) is found in [Labelle].
EXAMPLES:
The virtual species \(\Omega\) 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) doctest:...: DeprecationWarning: CombinatorialLogarithmSeries is deprecated, use CycleIndexSeriesRing(R).logarithm_series() or CycleIndexSeries().logarithm() instead See http://trac.sagemath.org/14846 for details. [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 \(\Omega \circ E^{+} = E^{+} \circ \Omega = X\) (that is, that composition with \(E^{+}\) in both directions yields the multiplicative identity \(X\)):
sage: Eplus = sage.combinat.species.set_species.SetSpecies(min=1).cycle_index_series() sage: CombinatorialLogarithmSeries().compose(Eplus).coefficients(4) [0, p[1], 0, 0]