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A combinatorial proof of the Harer-Zagier formula. (Démonstration combinatoire de la formule de Harer-Zagier.) (French. Abridged English version) Zbl 0984.05006

Summary: We give a combinatorial and self-contained proof of the Harer-Zagier formula for the numbers \(\varepsilon_g(m)\) of ways of obtaining a Riemann surface of given genus \(g\) by identifying in pairs the sides of a \(2m\)-gon. This formula was the key combinatorial fact needed for the calculation of the Euler characteristic of the moduli space of curves of genus \(g\). The method developed here completes the original combinatorial approach imagined by Harer and Zagier and avoids using the integration over a Gaussian ensemble of random matrices. Our derivation is based upon the enumeration of arborescences and Euler circuits.

MSC:

05A15 Exact enumeration problems, generating functions
14H10 Families, moduli of curves (algebraic)
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