arXiv subjects

Jacob A White

Publications and source records attributed to Jacob A White.

4 recordsLinked to original sources

arXiv2026

Cyclic Sieving Phenomenon for Independent sets of graphs

In this paper, we present examples of the cyclic sieving phenomenon coming from studying independent sets in graphs of a fixed size k. Given a graph G, and a cyclic group C acting on the graph, then C also acts on the collection of independent sets of G of a fixed size k. We exhibit cyclic sieving phenomena for a cyclic group acting on the collection of independent sets of powers of cycle graphs. As a corollary, we also find a closed formula for the number of independent sets of a given size in the power of a cycle graph, and in the power of a path. We also show how the graph construction of whiskering can be used to obtain new cyclic sieving phenomena from old phenomena. We also discuss recursive techniques to exhibit cyclic sieving phenomena for the independent sets of gear graphs, helm graphs, and book graphs.

math.CO↗

arXiv2021

The Chromatic Quasisymmetric Class Function of a Digraph

We introduce a quasisymmetric class function associated with a group acting on a double poset or on a directed graph. The latter is a generalization of the chromatic quasisymmetric function of a digraph introduced by Ellzey, while the latter is a generalization of a quasisymmetric function introduced by Grinberg. We prove representation-theoretic analogues of classical and recent results, including $F$-positivity, and combinatorial reciprocity theorems. We also deduce results for orbital quasisymmetric functions. We also study a generalization of the notion of strongly flawless sequences.

math.CO↗

arXiv2018

The Hopf monoid of Megagreedoids

We introduce megagreedoids, which generalize polymatroids, megamatroids, and greedoids. We define a quasisymmetric function invariant for a megagreedoid, and show that it has a positive expansion in the basis of fundamental quasisymmetric functions. Our proof involves lexicographic shellability. We also show that megagreedoids form a Hopf monoid. A running example is a megagreedoid associated to a rooted connected graph, and the resulting generalization of the chromatic symmetric function.

math.CO↗

arXiv2010

On Multivariate Chromatic Polynomials of Hypergraphs and Hyperedge Elimination

In this paper, we consider multivariate hyperedge elimination polynomials and multivariate chromatic polynomials for hypergraphs. The first set of polynomials is defined in terms of a deletion-contraction-extraction recurrence, previously investigated for graphs by Averbouch, Godlin, and Makowsky. The multivariate chromatic polynomial is an equivalent polynomial defined in terms of colorings, and generalizes the coboundary polynomial of Crapo, and the bivariate chromatic polynomial of Dohmen, Pönitz and Tittman. We show that specializations of these new polynomials recover polynomials which enumerate hyperedge coverings, matchings, transversals, and section hypergraphs. We also prove that the polynomials can be defined in terms of Möbius inversion on the bond lattice of a hypergraph, as well as compute these polynomials for various classes of hypergraphs.

math.CO↗