On the distribution of the number of cycles of elements in symmetric groups

Abstract

Summary: We give a formula for the number of elements in a fixed conjugacy class of a symmetric group whose product with a cyclic permutation has a given number of cycles. A consequence is a very short proof of the formula for the number \\\\varepsilong(n) of ways of obtaining a Riemann surface of given genus g by identifying in pairs the sides of a 2n-gon. This formula, originally proved by a considerably more difficult method in \\\\it J. Harer and the author [Invent. Math. 85, 457-485 (1986; Zbl 0616.14017)], was the key combinatorial fact needed there for the calculation of the Euler characteristic of the moduli space of curves of genus g. As a second application, we show that the number of ways of writing an even permutation π\\\\in ≥rm SN as product of two N-cycles always lies between 2(N- 1)!/(N- r+ 2) and 2(N- 1)!/(N- r+ 19/29), where r is the number of fixed points of π, and that both constants ``2'' and ``19/29'' are best possible.