Box polynomials and the excedance matrix

TL;DR

This paper studies box polynomials related to integer partitions within a grid, revealing their connections to combinatorial identities, set partitions, and the excedance matrix, with implications for graph coloring and algebraic structures.

Contribution

It introduces a novel expression of box polynomials via finite differences, extends identities using new operators, and links these polynomials to the excedance matrix and algebraic frameworks.

Findings

Box polynomials can be expressed using finite difference operators.

New identities for set partition enumeration are derived.

Connections between box polynomials, excedance matrix, and algebraic structures are established.

Abstract

We consider properties of the box polynomials, a one variable polynomial defined over all integer partitions $\lambda$ whose Young diagrams fit in an $m$ by $n$ box. We show that these polynomials can be expressed by the finite difference operator applied to the power $x^{m+n}$. Evaluating box polynomials yields a variety of identities involving set partition enumeration. We extend the latter identities using restricted growth words and a new operator called the fast Fourier operator, and consider connections between set partition enumeration and the chromatic polynomial on graphs. We also give connections between the box polynomials and the excedance matrix, which encodes combinatorial data from a noncommutative quotient algebra motivated by the recurrence for the excedance set statistic on permutations.