Extensions of partial cyclic orders and consecutive coordinate polytopes

TL;DR

This paper introduces new polytopes related to cyclic orders, explores their volumes and Ehrhart polynomials, and connects these geometric objects to classical combinatorial sequences like Euler and Narayana numbers.

Contribution

It provides a novel geometric framework for understanding cyclic orders and their enumeration through polytopes and Ehrhart theory.

Findings

Normalized volumes count circular extensions of partial cyclic orders

Ehrhart h*-polynomials relate to descents in cyclic orders

Special cases recover Euler, Eulerian, and Narayana numbers

Abstract

We introduce several classes of polytopes contained in $[0,1]^n$ and cut out by inequalities involving sums of consecutive coordinates. We show that the normalized volumes of these polytopes enumerate circular extensions of certain partial cyclic orders. Among other things this gives a new point of view on a question popularized by Stanley. We also provide a combinatorial interpretation of the Ehrhart $h^*$-polynomials of some of these polytopes in terms of descents of total cyclic orders. The Euler numbers, the Eulerian numbers and the Narayana numbers appear as special cases.