# Bivariate Order Polynomials

## TL;DR

This paper introduces a bivariate order polynomial extending Stanley's order polynomial, with decomposition formulas, reciprocity theorems, and links to bivariate chromatic polynomials, inspired by graph coloring concepts.

## Contribution

It presents a novel bivariate order polynomial with decomposition, reciprocity, and graph coloring connections, expanding combinatorial polynomial theory.

## Key findings

- Derived decomposition formulas in terms of linear extensions
- Established a combinatorial reciprocity theorem
- Connected the new polynomial to bivariate chromatic polynomials

## Abstract

Motivated by Dohmen-P\"onitz-Tittmann's bivariate chromatic polynomial $\chi_G(x,y)$, which counts all $x$-colorings of a graph $G$ such that adjacent vertices get different colors if they are $\le y$, we introduce a bivarate version of Stanley's order polynomial, which counts order preserving maps from a given poset to a chain. Our results include decomposition formulas in terms of linear extensions, a combinatorial reciprocity theorem, and connections to bivariate chromatic polynomials.