Counting with two-level polynomials

TL;DR

This paper introduces and analyzes two-level polynomials, a class of combinatorial counting functions with polynomial behavior in one parameter and polynomial degree in another, unifying various counting problems.

Contribution

The paper defines two-level polynomials, establishes their algebraic properties, and proves many combinatorial counting functions are instances of this class.

Findings

Chromatic polynomials for various graph families are two-level polynomials.

Partition functions into fixed parts are two-level polynomials.

Counting non-attacking chess piece placements are two-level polynomials.

Abstract

We examine combinatorial counting functions with two parameters, $n$ and $q$. For fixed $q$, these functions are (quasi-)polynomial in $n$. As $q$ varies, the degree of this polynomial is itself polynomial in $q$, as are the leading coefficients. We carefully define these two-level polynomials, lay out their basic algebraic properties, and provide a schema for showing a function is a two-level polynomial. Using the schema, we prove that a variety of counting functions arising in different areas of combinatorics are two-level polynomials. These include chromatic polynomials for many infinite families of graphs, partitions of an integer into a given number of parts, placing non-attacking chess pieces on a board, Sidon sets, and Sheffer sequences (including binomial type and Appell sequences).