Guillaume Chapuy - 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) - https://doi.org/10.46298/dmtcs.2747
A new combinatorial identity for unicellular maps, via a direct bijective approach.Conference paper

Authors: Guillaume Chapuy 1

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.


Volume: DMTCS Proceedings vol. AK, 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009)
Section: Proceedings
Published on: January 1, 2009
Imported on: January 31, 2017
Keywords: Polygon gluings,combinatorial identity,bijection,[MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO],[INFO.INFO-DM]Computer Science [cs]/Discrete Mathematics [cs.DM]
Funding:
    Source : OpenAIRE Graph
  • Combinatorial methods, from enumerative topology to random discrete structures and compact data representations.; Funder: European Commission; Code: 208471

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