The combinatorics of interval-vector polytopes

TL;DR

This paper explores the geometric and combinatorial properties of specific interval vector polytopes, revealing connections to Catalan numbers and Pascal's triangle, and providing insights into their structure.

Contribution

It introduces three classes of interval vector polytopes and uncovers their intriguing geometric-combinatorial properties, including volume and face number formulas.

Findings

One class has volumes equal to Catalan numbers.

Another class's face numbers follow Pascal 3-triangle.

The study reveals rich combinatorial structures in these polytopes.

Abstract

An \emph{interval vector} is a $(0,1)$-vector in $\mathbb{R}^n$ for which all the 1's appear consecutively, and an \emph{interval-vector polytope} is the convex hull of a set of interval vectors in $\mathbb{R}^n$. We study three particular classes of interval vector polytopes which exhibit interesting geometric-combinatorial structures; e.g., one class has volumes equal to the Catalan numbers, whereas another class has face numbers given by the Pascal 3-triangle.