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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., Comment: To appear in Eur. J. Combin., doi:10.1016/j.ejc.2014.07.002

For graphs $G$ and $H$, an $H$-coloring of $G$ is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colorings encode graph theory notions such as independent sets and proper colorings, and are a natural setting for the study of hard-constraint models in statistical physics. We study the set of $H$-colorings of the even discrete torus ${\mathbb Z}^d_m$, the graph on vertex set ${0, ..., m-1}^d$ ($m$ even) with two strings adjacent if they differ by 1 (mod $m$) on one coordinate and agree on all others. This is a bipartite graph, with bipartition classes ${\mathcal E}$ and ${\mathcal O}$. In the case $m=2$ the even discrete torus is the discrete hypercube or Hamming cube $Q_d$, the usual nearest neighbor graph on ${0,1}^d$. We obtain, for any $H$ and fixed $m$, a structural characterization of the space of $H$-colorings of ${\mathbb Z}^d_m$. We show that it may be partitioned into an exceptional subset of negligible size (as $d$ grows) and a collection of subsets indexed by certain pairs $(A,B) \in V(H)^2$, with each $H$-coloring in the subset indexed by $(A,B)$ having all but a vanishing proportion of vertices from ${\mathcal E}$ mapped to vertices from $A$, and all but a vanishing proportion of vertices from ${\mathcal O}$ mapped to vertices from $B$. This implies a long-range correlation phenomenon for uniformly chosen $H$-colorings of ${\mathbb Z}^d_m$ with $m$ fixed and $d$ growing. Our proof proceeds through an analysis of the entropy of a uniformly chosen $H$-coloring, and extends an approach of Kahn, who had considered the special case of $m=2$ and $H$ a doubly infinite path. All our results generalize to a natural weighted model of $H$-colorings., Comment: 29 pages, some corrections and minor revisions from earlier version, this version to appear in Journal of Combinatorial Theory Series B

For graphs $G$ and $H$, an {\em $H$-colouring} of $G$ (or {\em homomorphism} from $G$ to $H$) is a function from the vertices of $G$ to the vertices of $H$ that preserves adjacency. $H$-colourings generalize such graph theory notions as proper colourings and independent sets. For a given $H$, $k \in V(H)$ and $G$ we consider the proportion of vertices of $G$ that get mapped to $k$ in a uniformly chosen $H$-colouring of $G$. Our main result concerns this quantity when $G$ is regular and bipartite. We find numbers $0 \leq a^-(k) \leq a^+(k) \leq 1$ with the property that for all such $G$, with high probability the proportion is between $a^-(k)$ and $a^+(k)$, and we give examples where these extremes are achieved. For many $H$ we have $a^-(k) = a^+(k)$ for all $k$ and so in these cases we obtain a quite precise description of the almost sure appearance of a randomly chosen $H$-colouring. As a corollary, we show that in a uniform proper $q$-colouring of a regular bipartite graph, if $q$ is even then with high probability every colour appears on a proportion close to $1/q$ of the vertices, while if $q$ is odd then with high probability every colour appears on at least a proportion close to $1/(q+1)$ of the vertices and at most a proportion close to $1/(q-1)$ of the vertices. Our results generalize to natural models of weighted $H$-colourings, and also to bipartite graphs which are sufficiently close to regular. As an application of this latter extension we describe the typical structure of $H$-colourings of graphs which are obtained from $n$-regular bipartite graphs by percolation, and we show that $p=1/n$ is a threshold function across which the typical structure changes. The approach is through entropy, and extends work of J. Kahn, who considered the size of a randomly chosen independent set of a regular bipartite graph., Comment: 27 pages, small revisions from previous version, this version appears in Journal of Combinatorial Theory Series B