The f-vector of the descent polytope

TL;DR

This paper provides a detailed analysis of the f-vector of descent polytopes, expressing it as a sum over subsets, and explores its maximization, generating functions, and Ehrhart polynomials.

Contribution

It introduces a new combinatorial expression for the f-vector of descent polytopes and derives generating functions and maximization conditions.

Findings

f-vector expressed as a sum over all subsets of [n-1]

f-vector maximized for the alternating set {1,3,5,...}

generating functions for f-polynomial and Ehrhart polynomials derived

Abstract

For a positive integer n and a subset S of [n-1], the descent polytope DP_S is the set of points x_1, ..., x_n in the n-dimensional unit cube [0,1]^n such that x_i >= x_{i+1} for i in S and x_i <= x_{i+1} otherwise. First, we express the f-vector of DP_S as a sum over all subsets of [n-1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5, ...}. We derive a generating function for the f-polynomial F_S(t) of DP_S, written as a formal power series in two non-commuting variables with coefficients in Z[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.