Order-Chain Polytopes

TL;DR

This paper introduces order-chain polytopes, a new class of polytopes formed by intersections of order and chain polytopes, exploring their properties and volume relations with descent statistics.

Contribution

It defines order-chain polytopes and investigates their integrality, type inheritance, and volume properties, linking them to descent statistic maximization.

Findings

Order-chain polytopes form a new class of polytopes from order and chain polytopes.

Volume analysis reveals connections to descent statistic maximization.

Conditions for integrality and type inheritance are established.

Abstract

Given two families $X$ and $Y$ of integral polytopes with nice combinatorial and algebraic properties, a natural way to generate new class of polytopes is to take the intersection $\mathcal{P}=\mathcal{P}_1\cap\mathcal{P}_2$, where $\mathcal{P}_1\in X$, $\mathcal{P}_2\in Y$. Two basic questions then arise: 1) when $\mathcal{P}$ is integral and 2) whether $\mathcal{P}$ inherits the "old type" from $\mathcal{P}_1, \mathcal{P}_2$ or has a "new type", that is, whether $\mathcal{P}$ is unimodularly equivalent to some polytope in $X\cup Y$ or not. In this paper, we focus on the families of order polytopes and chain polytopes and create a new class of polytopes following the above framework, which are named order-chain polytopes. In the study on their volumes, we discover a natural relation with Ehrenborg and Mahajan's results on maximizing descent statistics.