A Colorful Involution for the Generating Function for Signed Stirling Numbers of the First Kind

We show how the generating function for signed Stirling numbers of the first kind can be proved using the involution principle and a natural combinatorial interpretation based on cycle-colored permuations. We seek an involution-based proof of the generating function for signed Stirling numbers of the first kind, written here as ∑ k (−1)c(n, k)x = (−1)(x)(x− 1) · · · (x− n+ 1) where c(n, k) is the number of permutations of [n] with k cycles. The standard proof uses [2] an algebraic manipulation of the generating function for unsigned Stirling numbers of the first kind. Fix an unordered x-set A; for example a set of x letters or “colors”. For π ∈ Sn, let Kπ be the set of disjoint cycles of π (including any cycles of length one). Let Sn,A = {(π, f) : π ∈ Sn; f : Kπ → A} be the set of cycle-colored permutations of [n], where f is interpreted as a “coloring” of the cycles of π using the “colors” of A. (We follow [1] in using colored permutations). Further let Kπ(i) be the unique cycle of π containing i for any 1 6 i 6 n, and κ(π) = |Kπ| be the number of cycles of π. Note that ∑