Combinatorially interpreting generalized Stirling numbers

Abstract

The Stirling numbers of the second kind nk (counting the number of partitions of a set of size n into k non-empty classes) satisfy the relation (xD)nf(x)=∑k≥0nkxkDkf(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))∑k≥0Sw(k)xkDkf(x) defines a sequence (Sw(k))k of Stirling numbers (of the second kind) of w. Explicit expressions for, and identities satisfied by, the Sw(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 nk as a count of partitions. Specifically, we associate to each w a quasi-threshold graph Gw, and we show that Sw(k) enumerates partitions of the vertex set of Gw into classes that do not span an edge of Gw. We use our interpretation to re-derive a known explicit expression for Sw(k), and in the case w=(xsDs)n to find a new summation formula linking Sw(k) to ordinary Stirling numbers. We also explore a natural q-analog of our interpretation.In the case w=(xrD)n it is known that Sw(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.