Factorizations of large cycles in the symmetric group Original Research Article

The factorizations of an n-cycle of the symmetric group Sn into m permutations with prescribed cycle types α1,…,αm describe topological equivalence classes of one pole meromorphic functions on Riemann surfaces. This is one of the motivations for a vast literature on counting such factorizations. Their number, denoted by cα1,…,αm(n), is also known as a connection coefficient of the center of the algebra of the symmetric group, whose multiplicative structure it describes. The relation to Riemann surfaces induces the definition of a genus for factorizations. It turns out that this genus is fully determined by the cycle types α1,…,αm, and that it has a determinant influence on the complexity of computing connection coefficients. In this article, a new formula for cα1,…,αm(n) is given, that makes this influence of the genus explicit. Moreover, our formula is cancellation-free, thus contrasting with known formulae in terms of characters of the symmetric group. This feature allows us to derive non-trivial asymptotic estimates. Our results rely on combining classical methods of the theory of characters of the symmetric group with a combinatorial approach that was first introduced in the much simpler case m=2 by Goupil and Schaeffer.