Guillaume Chapuy
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A new combinatorial identity for unicellular maps, via a direct bijective approach.
dmtcs:2747 -
Discrete Mathematics & Theoretical Computer Science,
January 1, 2009,
DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)
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https://doi.org/10.46298/dmtcs.2747
A new combinatorial identity for unicellular maps, via a direct bijective approach.Conference paper
We give a bijective operation that relates unicellular maps of given genus to unicellular maps of lower genus, with distinguished vertices. This gives a new combinatorial identity relating the number ϵg(n) of unicellular maps of size n and genus g to the numbers ϵj(n)'s, for j<g. In particular for each g this enables to compute the closed-form formula for ϵg(n) much more easily than with other known identities, like the Harer-Zagier formula. From the combinatorial point of view, we give an explanation to the fact that ϵg(n)=Rg(n)Cat(n), where Cat(n) is the n-th Catalan number and Rg is a polynomial of degree 3g, with explicit interpretation.
Combinatorial methods, from enumerative topology to random discrete structures and compact data representations.; Funder: European Commission; Code: 208471
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