Order polynomials and P\'olya's enumeration theorem

TL;DR

This paper extends Pólya's enumeration theorem to partially ordered sets and order-preserving maps, providing new reciprocity results and applications to counting graph colorings under symmetry.

Contribution

It generalizes Pólya's theorem to posets and order-preserving maps, introducing reciprocity results and applying them to graph coloring enumeration.

Findings

Generalization of Pólya's theorem to posets

Reciprocity theorem for order-preserving maps

Application to counting graph colorings

Abstract

P\'olya's enumeration theorem is concerned with counting labeled sets up to symmetry. Given a finite group acting on a finite set of labeled elements it states that the number of labeled sets up to symmetry is given by a polynomial in the number of labels. We give a new perspective on this theorem by generalizing it to partially ordered sets and order preserving maps. Further we prove a reciprocity statement in terms of strictly order preserving maps generalizing a classical result by Stanley (1970). We apply our results to counting graph colorings up to symmetry.