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Enumeration of quadratic words of given genus

Counting the number of ways to obtain a surface by gluing the sides of a polygon, preprint (1992).


This was an exercise, but not an easy one. Harer and Zagier [HZ] proved that the virtual Euler characteristic of the mapping class group of the orientable closed surface of genus g is $\zeta(1-2g)/(2-2g)$. A crucial part of their proof is the recurrence formula

\begin{displaymath}(r+1)\varepsilon_g(r) = (4r-2)\varepsilon_g(r-1) +
(2r-1)(r-1)(2r-3)\varepsilon_{g-1}(r-2), \end{displaymath}

where $\varepsilon_g(n)$ is the number of ways to obtain an orientable surface of genus g by pairing the sides of a 2n-gon. Their proof is long and uses heavy machinery; they ask for an elementary one. My paper gives an elementary proof, though not very short.


J. Harer and D. Zagier, The Euler characteristic of the moduli space of curves, Invent. Math. 85 (1986), 457-485.