Polynomial sequences of binomial-type arising in graph theory

TL;DR

This paper demonstrates that the number of ways to tile an n×n toroidal chessboard with fixed polyominos forms polynomial sequences of binomial type, with applications to graph coloring and higher-dimensional lattices.

Contribution

It establishes that tiling counts are polynomial sequences of binomial type and extends this to higher dimensions and other lattices, also applying results to chromatic polynomial coefficients.

Findings

Tiling counts are polynomial in n for fixed polyominos.

These polynomials satisfy binomial-type recurrences.

Chromatic polynomial coefficients also follow binomial-type recurrences.

Abstract

In this paper, we show that the solution to a large class of "tiling" problems is given by a polynomial sequence of binomial type. More specifically, we show that the number of ways to place a fixed set of polyominos on an $n\times n$ toroidal chessboard such that no two polyominos overlap is eventually a polynomial in $n$, and that certain sets of these polynomials satisfy binomial-type recurrences. We exhibit generalizations of this theorem to higher dimensions and other lattices. Finally, we apply the techniques developed in this paper to resolve an open question about the structure of coefficients of chromatic polynomials of certain grid graphs (namely that they also satisfy a binomial-type recurrence).