Factorizations of large cycles in the symmetric group

Dominique Poulalhon, Gilles Schaeffer
2002 Discrete Mathematics  
The factorizations of an n-cycle of the symmetric group Sn into m permutations with prescribed cycle types 1; : : : ; m 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 c (n) 1 ;:::; m , is also known as a connection coe cient of the center of the algebra of the symmetric group, whose multiplicative structure it describes. The relation to
more » ... iemann surfaces induces the deÿnition of a genus for factorizations. It turns out that this genus is fully determined by the cycle types 1; : : : ; m, and that it has a determinant in uence on the complexity of computing connection coe cients. In this article, a new formula for c (n) 1 ;:::; m is given, that makes this in uence 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 ÿrst introduced in the much simpler case m = 2 by Goupil and Schae er.
doi:10.1016/s0012-365x(01)00361-2 fatcat:yeummsfaevhazamiu2koskcczy