On the number of factorizations of a full cycle

i mi!. A hook is a partition of the form [1b, a + 1] with a, b ≥ 0. We use Frobenius notation for hooks, writing (a|b) in place of [1b, a+ 1]. The conjugacy class of the symmetric group Sn consisting of all n!/zλ permutations of cycle type λ ⊢ n will be denoted by Cλ. The irreducible characters χλ of Sn are naturally indexed by partitions λ of n, and we use the usual notation χμ for the common value of χ λ at any element of Cμ. We write f λ for the degree χ[1n] of χ λ. For vectors j = (j1, . . . , jm) and x = (x1, . . . , xm) we use the abbreviations j! = j1! · · · jm! and x = x1 1 · · · x jm m . Finally, if α ∈ Q and f ∈ Q[[x]] is a formal power series, then we write [αx] f(x) for the coefficient of the monomial αx in f(x).