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:

\[\Omega \circ E^{+} = E^{+} \circ \Omega = X.\]

AUTHORS:

  • Andrew Gainer-Dewar (2013): initial version

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]
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)
[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]

REFERENCES:

[BLL]
  1. Bergeron, G. Labelle, and P. Leroux. “Combinatorial species and tree-like structures”. Encyclopedia of Mathematics and its Applications, vol. 67, Cambridge Univ. Press. 1998.
[Labelle]
  1. Labelle. “New combinatorial computational methods arising from pseudo-singletons.” DMTCS Proceedings 1, 2008.

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