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Enumeration of quadratic words of given genus

1.
Counting the number of ways to obtain a surface by gluing the sides of a polygon, preprint (1992).



SUMMARY:


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.



REFERENCES:


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




2000-03-16