Discrete Mathematics & Theoretical Computer Science (Jan 2009)
A new combinatorial identity for unicellular maps, via a direct bijective approach.
Abstract
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 $\epsilon_g(n)$ of unicellular maps of size $n$ and genus $g$ to the numbers $\epsilon _j(n)$'s, for $j \lt g$. In particular for each $g$ this enables to compute the closed-form formula for $\epsilon_g(n)$ 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 $\epsilon_g(n)=R_g(n) \mathrm{Cat}(n)$, where $\mathrm{Cat}(n$) is the $n$-th Catalan number and $R_g$ is a polynomial of degree $3g$, with explicit interpretation.
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