The f-Vector of the Descent Polytope

Abstract

For a positive integer n and a subset S⊆[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 if i∈S and x i ≤x i+1 otherwise. First, we express the f-vector 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,…}∩[n−1]. We derive a generating function for F S (t), 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.