Order and chain polytopes of maximal ranked posets

TL;DR

This paper proves a stronger version of a conjecture relating the $f$-vectors of order and chain polytopes for a special class of posets, showing monotonic increase over certain families.

Contribution

It establishes that for a specific class of posets, the $f$-vectors of chain and order polytopes increase monotonically within admissible families.

Findings

The $f$-vector of the chain polytope dominates that of the order polytope.

Monotonic increase of $f$-vectors is proven for a special class of posets.

A stronger form of Hibi and Li's conjecture is demonstrated.

Abstract

The order and chain polytopes, introduced by Richard P. Stanley, form a pair of Ehrhart equivalent polytopes associated to a given finite poset. A conjecture by Takayuki Hibi and Nan Li states that the $f$-vector of the chain polytope dominates the $f$-vector of the order polytope. In this paper we prove a stronger form of that conjecture for a special class of posets. More precisely, we show that the $f$-vectors increase monotonically over an admissible family of chain-order polytopes for such posets.