Combinatorially interpreting generalized Stirling numbers

The Stirling numbers of the second kind {(n)(k)} (counting the number of partitions of a set of size n into k non-empty classes) satisfy the relation (xD)(n) f (x) = Sigma(k >= 0) {(n)(k)}x(k)D(k)f(x) where f is an arbitrary function and D is differentiation with respect to x. More generally, for every word w in alphabet {x, D} the identity wf (x) = x((#(x's in w)-#(D's in w))) Sigma S-k >= 0(w)(k)x(k)D(k)f(x) defines a sequence (S-w(k))(k) of Stirling numbers (of the second kind) of w. Explicit expressions for, and identities satisfied by, the S-w(k) have been obtained by numerous authors, and combinatorial interpretations have been presented. Here we provide a new combinatorial interpretation that, unlike previous ones, retains the spirit of the familiar interpretation of {(n)(k)} as a count of partitions. Specifically, we associate to each w a quasi-threshold graph G(w), and we show that S-w(k) enumerates partitions of the vertex set of G(w), into classes that do not span an edge of G(w). We use our interpretation to re-derive a known explicit expression for S-w(k), and in the case w = (x(s)D(s))(n) to find a new summation formula linking S-w(k) to ordinary Stirling numbers. We also explore a natural q-analog of our interpretation. In the case w = (x(r)D)(n) it is known that S-w(k) counts increasing, n-vertex, k-component r-ary forests. Motivated by our combinatorial interpretation we exhibit bijections between increasing r-ary forests and certain classes of restricted partitions. (C) 2014 Elsevier Ltd. All rights reserved.