The Stirling numbers of the second kind {nk} (counting the number of partitions of a set of size nn into kk non-empty classes) satisfy the relation (xD)nf(x)=∑k≥0{nk}xkDkf(x) Turn MathJax on

where ff is an arbitrary function and DD is differentiation with respect to xx. More generally, for every word ww in alphabet {x,D}{x,D}the identity wf(x)=x(#(x’s in w)−#(D’s in w))∑k≥0Sw(k)xkDkf(x) Turn MathJax on

defines a sequence (S_{w}(k))_{k}(Sw(k))k of *Stirling numbers (of the second kind) * of ww. Explicit expressions for, and identities satisfied by, the S_{w}(k)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 ww a quasi-threshold graph G_{w}Gw, and we show that S_{w}(k)Sw(k) enumerates partitions of the vertex set of G_{w}Gw into classes that do not span an edge of G_{w}Gw. We use our interpretation to re-derive a known explicit expression for S_{w}(k)Sw(k), and in the case w=(x^{s}D^{s})^{n}w=(xsDs)n to find a new summation formula linking S_{w}(k)Sw(k) to ordinary Stirling numbers. We also explore a natural qq-analog of our interpretation.

In the case w=(x^{r}D)^{n}w=(xrD)n it is known that S_{w}(k)Sw(k) counts increasing, nn-vertex, kk-component rr-ary forests. Motivated by our combinatorial interpretation we exhibit bijections between increasing rr-ary forests and certain classes of restricted partitions.