Combinatorial Reciprocity Theorems

TL;DR

This paper explores combinatorial reciprocity theorems by examining how counting functions related to hyperplane arrangements, lattice points, graph colorings, and P-partitions reveal interesting properties when evaluated at negative integers, linking geometry and combinatorics.

Contribution

It provides a unifying geometric perspective on various combinatorial reciprocity theorems involving counting functions evaluated at negative integers.

Findings

Counting functions reveal new combinatorial insights at negative integers.

A geometric framework unifies different reciprocity theorems.

Evaluation at negative integers often yields meaningful combinatorial interpretations.

Abstract

A common theme of enumerative combinatorics is formed by counting functions that are polynomials evaluated at positive integers. In this expository paper, we focus on four families of such counting functions connected to hyperplane arrangements, lattice points in polyhedra, proper colorings of graphs, and $P$-partitions. We will see that in each instance we get interesting information out of a counting function when we evaluate it at a \emph{negative} integer (and so, a priori the counting function does not make sense at this number). Our goals are to convey some of the charm these "alternative" evaluations of counting functions exhibit, and to weave a unifying thread through various combinatorial reciprocity theorems by looking at them through the lens of geometry, which will include some scenic detours through other combinatorial concepts.