  Let $w$ be a word in alphabet $\{x,D\}$ with $m$ $x$'s and $n$ $D$'s.
Interpreting "$x$" as multiplication by $x$, and "$D$" as differentiation with
respect to $x$, the identity $wf(x) = x^{m-n}\sum_k S_w(k) x^k D^k f(x)$, valid
for any smooth function $f(x)$, defines a sequence $(S_w(k))_k$, the terms of
which we refer to as the {\em Stirling numbers (of the second kind)} of $w$.
The nomenclature comes from the fact that when $w=(xD)^n$, we have $S_w(k)={n
\brace k}$, the ordinary Stirling number of the second kind.
  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 retains the
spirit of the familiar interpretation of ${n \brace k}$ 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 also discuss some
relatives of, and consequences of, our interpretation, including $q$-analogs
and bijections between families of labelled forests and sets of restricted
partitions.
