Euler characteristic reciprocity for chromatic, flow and order polynomials

TL;DR

This paper extends reciprocity theorems for chromatic, flow, and order polynomials by employing Euler characteristics of semialgebraic sets and introducing semialgebraic posets, allowing for negative set concepts.

Contribution

It introduces semialgebraic posets and establishes Euler characteristic reciprocity theorems for various polynomials, generalizing classical combinatorial results.

Findings

Euler characteristic reciprocity for order polynomials

Euler characteristic reciprocity for chromatic polynomials

Euler characteristic reciprocity for flow polynomials

Abstract

The Euler characteristic of a semialgebraic set can be considered as a generalization of the cardinality of a finite set. An advantage of semialgebraic sets is that we can define "negative sets" to be the sets with negative Euler characteristics. Applying this idea to posets, we introduce the notion of semialgebraic posets. Using "negative posets", we establish Stanley's reciprocity theorems for order polynomials at the level of Euler characteristics. We also formulate the Euler characteristic reciprocities for chromatic and flow polynomials.