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# 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] | 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] | Labelle. “New combinatorial computational methods arising from pseudo-singletons.” DMTCS Proceedings 1, 2008.

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