Counting orbits of a product of permutations (Q1813986)

Let \(\sigma\) and \(\tau\) be permutations on \(n\) letters. It is shown that the numbers of orbits of \(\sigma\), \(\tau\) and \(\sigma\tau\) are related to the Euler characteristic of an orientable surface \(S_{\sigma,\tau}\) that can be naturally assigned to \(\sigma\) and \(\tau\). The Euler formula is then used to express the number of orbits of \(\sigma\tau\) by means of the rank of the matrix corresponding to the cup product bilinear form on the first cohomology group of \(S_{\sigma,\tau}\).