Arithmetic of marked order polytopes, monotone triangle reciprocity, and partial colorings

TL;DR

This paper studies the properties of marked order polytopes, proving they have piecewise polynomial counting functions and reciprocity relations, and applies these results to monotone triangle enumeration and partial graph coloring extensions.

Contribution

It introduces a new geometric framework for counting integral extensions of order-preserving maps and establishes reciprocity theorems, connecting combinatorics and polyhedral geometry.

Findings

Counting functions are piecewise polynomial in F

Reciprocity relations are established for order-reversing maps

Provides geometric proofs for known combinatorial reciprocity results

Abstract

For a poset P, a subposet A, and an order preserving map F from A into the real numbers, the marked order polytope parametrizes the order preserving extensions of F to P. We show that the function counting integral-valued extensions is a piecewise polynomial in F and we prove a reciprocity statement in terms of order-reversing maps. We apply our results to give a geometric proof of a combinatorial reciprocity for monotone triangles due to Fischer and Riegler (2011) and we consider the enumerative problem of counting extensions of partial graph colorings of Herzberg and Murty (2007).