Arithmetic of Marked Order Polytopes, Monotone Triangle Reciprocity, and Partial Colorings

Abstract

For a pair of posets $A \subseteq P$ and an order preserving map $\lambda : A \rightarrow \mathbb{R}$, the marked order polytope parametrizes the order preserving extensions of $\lambda$ to $P$. We show that the function counting integer-valued extensions is a piecewise polynomial in $\lambda$ 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 [J. Combin. Theory Ser. A, 120 (2013), pp. 1372--1393] and we consider the enumerative problem of counting extensions of partial graph colorings of Herzberg and Murty [Notices Amer. Math. Soc., 54 (2007), pp. 708--717].