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The reciprocal of $e^{-x}$ has a power series about $0$ in which all coefficients are non-negative. Gessel [Reciprocals of exponential polynomials and permutation enumeration, Australas. J. Combin., 74, 2019] considered truncates of the power series of $e^{-x}$, i.e. polynomials of the form $\sum_{n=0}^r (-1)^n\frac{x^n}{n!}$, and established combinatorially that the reciprocal of the truncate has a power series with all coefficients non-negative precisely when $r$ is odd. Here we extend Gessel's observations to arbitrary ``thinned exponential series''. To be precise, let $A \subseteq \{1,3,5,\ldots\}$ and $B \subseteq \{2,4,6,\ldots\}$, and consider the series \[ 1-\sum_{a \in A} \frac{x^a}{a!} + \sum_{b \in B} \frac{x^b}{b!}. \] We consider conditions on $A$ and $B$ that ensure that the reciprocal series has all coefficients non-negative. We give combinatorial proofs for a large set of conditions, including whenever $1 \in A$ and the endpoints of the maximal consecutive intervals in $A \cup B$ are odd integers. In particular, the coefficients in the reciprocal series can be interpreted as ordered set partitions of $[n]$ with block size restrictions, or in terms of permutations with restricted lengths of maximally increasing runs, suitably weighted., Comment: 34 pages, revised, a new section 6 proves the main result in an alternative way using the Run Theorem

Let $P_G(k)$ be the number of proper $k$-colorings of a finite simple graph $G$. Tomescu's conjecture, which was recently solved by Fox, He, and Manners, states that $P_G(k) \le k!(k-1)^{n-k}$ for all connected graphs $G$ on $n$ vertices with chromatic number $k\geq 4$. In this paper, we study the same problem with the additional constraint that $G$ is $\ell$-connected. For $2$-connected graphs $G$, we prove a tight bound \[ P_G(k) \le (k-1)!((k-1)^{n-k+1} + (-1)^{n-k}), \] and show that equality is only achieved if $G$ is a $k$-clique with an ear attached. For $\ell \ge 3$, we prove an asymptotically tight upper bound \[ P_G(k) \le k!(k-1)^{n-\ell - k + 1} + O((k-2)^n), \] and provide a matching lower bound construction. For the ranges $k \geq \ell$ or $\ell \geq (k-2)(k-1)+1$ we further find the unique graph maximizing $P_G(k)$. We also consider generalizing $\ell$-connected graphs to connected graphs with minimum degree $\delta$.

Worpitzky's identity expresses $n^p$ in terms of the Eulerian numbers and binomial coefficients: $$n^p = \sum_{i=0}^{p-1} \genfrac<>{0pt}{}{p}{i} \binom{n+i}{p}.$$ Pita-Ruiz recently defined numbers $A_{a,b,r}(p,i)$ implicitly to satisfy a generalized Worpitzky identity $$\binom{an+b}{r}^p = \sum_{i=0}^{rp} A_{a,b,r}(p,i) \binom{n+rp-i}{rp},$$ and asked whether there is a combinatorial interpretation of the numbers $A_{a,b,r}(p,i)$. We provide such a combinatorial interpretation by defining a notion of descents in colored multipermutations, and then proving that $A_{a,b,r}(p,i)$ is equal to the number of colored multipermutations of $\{1^r, 2^r, \ldots, p^r\}$ with $a$ colors and $i$ weak descents. We use this to give combinatorial proofs of several identities involving $A_{a,b,r}(p,i)$, including the aforementioned generalized Worpitzky identity.

We study the problem of maximizing the number of independent sets in $n$-vertex $k$-chromatic $\ell$-connected graphs. First we consider maximizing the total number of independent sets in such graphs with $n$ sufficiently large, and for this problem we use a stability argument to find the unique extremal graph. We show that our result holds within the larger family of $n$-vertex $k$-chromatic graphs with minimum degree at least $\ell$, again for $n$ sufficiently large. We also maximize the number of independent sets of each fixed size in $n$-vertex 3-chromatic 2-connected graphs. We finally address maximizing the number of independent sets of size 2 (equivalently, minimizing the number of edges) over all $n$-vertex $k$-chromatic $\ell$-connected graphs.

We classify the trees on $n$ vertices with the maximum and the minimum number of certain generalized colorings, including conflict-free, odd, non-monochromatic, star, and star rainbow vertex colorings. We also extend a result of Cutler and Radcliffe on the maximum and minimum number of existence homomorphisms from a tree to a completely looped graph on $q$ vertices., Comment: 12 pages

Given $R \subseteq \mathbb{N}$ let ${n \brace k}_R$, ${n \brack k}_R$, and $L(n,k)_R$ be the number of ways of partitioning the set $[n]$ into $k$ non-empty subsets, cycles and lists, respectively, with each block having cardinality in $R$. We refer to these as the $R$-restricted Stirling numbers of the second and first kind and the $R$-restricted Lah numbers, respectively. Note that the classical Stirling numbers of the second kind and first kind, and Lah numbers are ${n \brace k} = {n \brace k}_{\mathbb{N}}$, ${n \brack k} = {n \brack k}_{\mathbb{N}} $ and $L(n,k) = L(n,k)_{\mathbb{N}}$, respectively. The matrices $[{n \brace k}]_{n,k \geq 1}$, $[{n \brack k}]_{n,k \geq 1}$ and $[L(n,k)]_{n,k \geq 1}$ have inverses $[(-1)^{n-k}{n \brack k}]_{n,k \geq 1}$, $[(-1)^{n-k} {n \brace k}]_{n,k \geq 1}$ and $[(-1)^{n-k} L(n,k)]_{n,k \geq 1}$ respectively. The inverse matrices $[{n \brace k}_R]^{-1}_{n,k \geq 1}$, $[{n \brack k}_R]^{-1}_{n,k \geq 1}$ and $[L(n,k)_R]^{-1}_{n,k \geq 1}$ exist if and only if $1 \in R$. We express each entry of each of these matrices as the difference between the cardinalities of two explicitly defined families of labeled forests. In particular the entries of $[{n \brace k}_{[r]}]^{-1}_{n,k \geq 1}$ have combinatorial interpretations, affirmatively answering a question of Choi, Long, Ng and Smith from 2006. If $1,2 \in R$ and if for all $n \in R$ with $n$ odd and $n \geq 3$, we have $n \pm 1 \in R$, we additionally show that each entry of $[{n \brace k}_R]^{-1}_{n,k \geq 1}$, $[{n \brack k}_R]^{-1}_{n,k \geq 1}$ and $[L(n,k)_R]^{-1}_{n,k \geq 1}$ is up to an explicit sign the cardinality of a single explicitly defined family of labeled forests. Our results also provide combinatorial interpretations of the $k$th Whitney numbers of the first and second kinds of $\Pi_n^{1,d}$, the poset of partitions of $[n]$ that have each part size congruent to $1$ mod $d$., Comment: This is a substantial revision of version 1, with more extensive results and unified proofs, as well as with new connections to certain Whitney numbers

For graphs $G$ and $H$, an $H$-coloring of $G$ is a map from the vertices of $G$ to the vertices of $H$ that preserves edge adjacency. We consider the following extremal enumerative question: for a given $H$, which connected $n$-vertex graph with minimum degree $\delta$ maximizes the number of $H$-colorings? We show that for non-regular $H$ and sufficiently large $n$, the complete bipartite graph $K_{\delta,n-\delta}$ is the unique maximizer. As a corollary, for non-regular $H$ and sufficiently large $n$ the graph $K_{k,n-k}$ is the unique $k$-connected graph that maximizes the number of $H$-colorings among all $k$-connected graphs. Finally, we show that this conclusion does not hold for all regular $H$ by exhibiting a connected $n$-vertex graph with minimum degree $\delta$ which has more $K_{q}$-colorings (for sufficiently large $q$ and $n$) than $K_{\delta,n-\delta}$., Comment: 10 pages, to appear in Journal of Graph Theory

Let $\mathfrak{S}_n$ and $\mathfrak{B}_n$ denote the respective sets of ordinary and bigrassmannian (BG) permutations of order $n$, and let $(\mathfrak{S}_n,\leq)$ denote the Bruhat ordering permutation poset. We study the restricted poset $(\mathfrak{B}_n,\leq)$, first providing a simple criterion for comparability. This criterion is used to show that that the poset is connected, to enumerate the saturated chains between elements, and to enumerate the number of maximal elements below $r$ fixed elements. It also quickly produces formulas for $\beta(\omega)$ ($\alpha(\omega)$ respectively), the number of BG permutations weakly below (weakly above respectively) a fixed $\omega\in\mathfrak{B}_n$, and is used to compute the M\"obius function on any interval in $\mathfrak{B}_n$. We then turn to a probabilistic study of $\beta=\beta(\omega)$ ($\alpha=\alpha(\omega)$ respectively) for the uniformly random $\omega\in\mathfrak{B}_n$. We show that $\alpha$ and $\beta$ are equidistributed, and that $\beta$ is of the same order as its expectation with high probability, but fails to concentrate about its mean. This latter fact derives from the limiting distribution of $\beta/n^3$. We also compute the probability that randomly chosen BG permutations form a 2- or 3-element multichain., Comment: 36 pages. This version is a significant edit and extension of the previous version. It includes a new title, a computation of the Mobius function, and a probabilistic study of the poset. This version to appear in the Australasian Journal of Combinatorics

For graphs $G$ and $H$, an $H$-coloring of $G$ is an adjacency preserving map from the vertices of $G$ to the vertices of $H$. $H$-colorings generalize such notions as independent sets and proper colorings in graphs. There has been much recent research on the extremal question of finding the graph(s) among a fixed family that maximize or minimize the number of $H$-colorings. In this paper, we prove several results in this area. First, we find a class of graphs ${\mathcal H}$ with the property that for each $H \in {\mathcal H}$, the $n$-vertex tree that minimizes the number of $H$-colorings is the path $P_n$. We then present a new proof of a theorem of Sidorenko, valid for large $n$, that for every $H$ the star $K_{1,n-1}$ is the $n$-vertex tree that maximizes the number of $H$-colorings. Our proof uses a stability technique which we also use to show that for any non-regular $H$ (and certain regular $H$) the complete bipartite graph $K_{2,n-2}$ maximizes the number of $H$-colorings of $n$-vertex $2$-connected graphs. Finally, we show that the cycle $C_n$ maximizes the number of proper colorings of $n$-vertex $2$-connected graphs., Comment: 14 pages, to appear in JCTB

The aim of this paper is to discuss potential advances in PET kinetic models and direct reconstruction of kinetic parameters. As a prominent example we focus on a typical task in perfusion imaging and derive a system of transport-reaction-diffusion equations, which is able to include macroscopic flow properties in addition to the usual exchange between arteries, veins, and tissues. For this system we propose an inverse problem of estimating all relevant parameters from PET data. We interpret the parameter identification as a nonlinear inverse problem, for which we formulate and analyze variational regularization approaches. For the numerical solution we employ gradient-based methods and appropriate splitting methods, which are used to investigate some test cases.

The game of peg solitaire on graphs was introduced by Beeler and Hoilman in 2011. In this game, pegs are initially placed on all but one vertex of a graph $G$. If $xyz$ forms a path in $G$ and there are pegs on vertices $x$ and $y$ but not $z$, then a {\em jump} places a peg on $z$ and removes the pegs from $x$ and $y$. A graph is called solvable if, for some configuration of pegs occupying all but one vertex, some sequence of jumps leaves a single peg. We study the game of {\em reversible peg solitaire}, where there are again initially pegs on all but one vertex, but now both jumps and unjumps (the reversal of a jump) are allowed. We show that in this game all non-star graphs that contain a vertex of degree at least three are solvable, that cycles and paths on $n$ vertices, where $n$ is divisible by $2$ or $3$, are solvable, and that all other graphs are not solvable. We also classify the possible starting hole and ending peg positions for solvable graphs., Comment: 9 pages, 4 figures. This version includes a title change, minor edits based on referee comments, and a classification of starting hole and ending peg locations. To appear in Discrete Math

The lattice of monotone triangles $(\mathfrak{M}_n,\le)$ ordered by entry-wise comparisons is studied. Let $\tau_{\min}$ denote the unique minimal element in this lattice, and $\tau_{\max}$ the unique maximum. The number of $r$-tuples of monotone triangles $(\tau_1,\ldots,\tau_r)$ with minimal infimum $\tau_{\min}$ (maximal supremum $\tau_{\max}$, resp.) is shown to asymptotically approach $r|\mathfrak{M}_n|^{r-1}$ as $n \to \infty$. Thus, with high probability this event implies that one of the $\tau_i$ is $\tau_{\min}$ ($\tau_{\max}$, resp.). Higher-order error terms are also discussed., Comment: 15 pages

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 $H$, a homomorphism from $G$ to $H$, or $H$-coloring of $G$, is a map from the vertices of $G$ to the vertices of $H$ that preserves adjacency. When $H$ is composed of an edge with one looped endvertex, an $H$-coloring of $G$ corresponds to an independent set in $G$. Galvin showed that, for sufficiently large $n$, the complete bipartite graph $K_{\delta,n-\delta}$ is the $n$-vertex graph with minimum degree $\delta$ that has the largest number of independent sets. In this paper, we begin the project of generalizing this result to arbitrary $H$. Writing $\hom(G,H)$ for the number of $H$-colorings of $G$, we show that for fixed $H$ and $\delta = 1$ or $\delta = 2$, \[ \hom(G,H) \leq \max \{\hom(K_{\delta+1},H)^{\frac{n}{\delta+1}}, \hom(K_{\delta,\delta},H)^{\frac{n}{2\delta}}, \hom(K_{\delta,n-\delta},H)\} \] for any $n$-vertex $G$ with minimum degree $\delta$ (for sufficiently large $n$). We also provide examples of $H$ for which the maximum is achieved by $\hom(K_{\delta+1},H)^{\frac{n}{\delta+1}}$ and other $H$ for which the maximum is achieved by $\hom(K_{\delta,\delta},H)^{\frac{n}{2\delta}}$. For $\delta \geq 3$ (and sufficiently large $n$), we provide a infinite family of $H$ for which $\hom(G,H) \leq \hom(K_{\delta,n-\delta},H)$ for any $n$-vertex $G$ with minimum degree $\delta$. The results generalize to weighted $H$-colorings., Comment: 26 pages, 5 figures, final version

Galvin showed that for all fixed $\delta$ and sufficiently large $n$, the $n$-vertex graph with minimum degree $\delta$ that admits the most independent sets is the complete bipartite graph $K_{\delta,n-\delta}$. He conjectured that except perhaps for some small values of $t$, the same graph yields the maximum count of independent sets of size $t$ for each possible $t$. Evidence for this conjecture was recently provided by Alexander, Cutler, and Mink, who showed that for all triples $(n,\delta, t)$ with $t\geq 3$, no $n$-vertex {\em bipartite} graph with minimum degree $\delta$ admits more independent sets of size $t$ than $K_{\delta,n-\delta}$. Here we make further progress. We show that for all triples $(n,\delta,t)$ with $\delta \leq 3$ and $t\geq 3$, no $n$-vertex graph with minimum degree $\delta$ admits more independent sets of size $t$ than $K_{\delta,n-\delta}$, and we obtain the same conclusion for $\delta > 3$ and $t \geq 2\delta +1$. Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree $\delta$ whose minimum degree drops on deletion of an edge or a vertex., Comment: 21 pages, 6 figures

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