John Engbers

Combinatorially interpreting generalized Stirling numbers

The Stirling numbers of the second kind n k (counting the number of partitions of a set of size n into k non-empty classes) satisfy the relation ( x D ) n f ( x ) = ? k ? 0 n k x k D k f ( 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 w f ( x ) = x ( # ( x s?in? w ) - # ( D s?in? w ) ) ? k ? 0 S w ( k ) x k D k f ( x ) defines a sequence ( S w ( k ) ) k of Stirling numbers (of the second kind) of w . 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, unlike previous ones, retains the spirit of the familiar interpretation of n 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 use our interpretation to re-derive a known explicit expression for S w ( k ) , and in the case w = ( x s D s ) n to find a new summation formula linking S w ( k ) to ordinary Stirling numbers. We also explore a natural q -analog of our interpretation.In the case w = ( x r D ) n it is known that S w ( 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.

Counting Independent Sets of a Fixed Size in Graphs with a Given Minimum Degree

Galvin showed that for all fixed δ and sufficiently large n, the n-vertex graph with minimum degree δ that admits the most independent sets is the complete bipartite graph . 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 with , no n-vertex bipartite graph with minimum degree δ admits more independent sets of size t than . Here, we make further progress. We show that for all triples with and , no n-vertex graph with minimum degree δ admits more independent sets of size t than , and we obtain the same conclusion for and . Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree δ whose minimum degree drops on deletion of an edge or a vertex.

Extremal H-colorings of trees and 2-connected graphs

Abstract 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 H with the property that for each H ∈ 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 has the most proper q -colorings among all n -vertex 2-connected graphs.

Extremal H-Colorings of Graphs with Fixed Minimum Degree

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 Ki¾?,n-i¾? is the n-vertex graph with minimum degree i¾? that has the largest number of independent sets. In this article, we begin the project of generalizing this result to arbitrary H. Writing homG,H for the number of H-colorings of G, we show that for fixed H and i¾?=1 or i¾?=2,homG,Hi¾?max{homKi¾?+1,Hni¾?+1,homKi¾?,i¾?,Hn2i¾?,homKi¾?,n-i¾?,H}for any n-vertex G with minimum degree i¾? for sufficiently large n. We also provide examples of H for which the maximum is achieved by homKi¾?+1,Hni¾?+1 and other H for which the maximum is achieved by homKi¾?,i¾?,Hn2i¾?. For i¾?i¾?3 and sufficiently large n, we provide an infinite family of H for which homG,Hi¾?homKi¾?,n-i¾?,H for any n-vertex G with minimum degree i¾?. The results generalize to weighted H-colorings.

Restricted Stirling and Lah number matrices and their inverses

Given

Maximizing H‐Colorings of Connected Graphs with Fixed Minimum Degree

R \subseteq \mathbb{N}

Reversible peg solitaire on graphs

let

A Direct Proof of the Integral Formulae for the Inverse Hyperbolic Functions

{n \brace k}_R

Trivial Meet and Join within the Lattice of Monotone Triangles

,

On Comparability of Bigrassmannian Permutations

{n \brack k}_R