Counting Cycles in Permutations by Group Characters, With an Application to a Topological Problem
David M. Jackson · Transactions of the American Mathematical Society · 1987
The character theory of the symmetric group is used to derive properties of the number of permutations, with $k$ cycles, which are expressible as the product of a full cycle with an element of an arbitrary, but fixed, conjugacy class. For the conjugacy class of fixed point free involutions, this problem has application to the analysis of singularities in surfaces.