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.

Random doubly stochastic tridiagonal matrices

“…See [19] for more on these Fibonacci permutations, including a study of the graph formed by the vertices and edges of the polytope \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document} n . Chebikin and Ehrenborg [9] give a nice but somewhat complicated expression for the generating function for the f ‐vector of \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document} n . See [18] for a combinatorial description of the faces of the polytope \documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath, amssymb}\pagestyle{empty}\begin{document}\begin{align*}{\mathcal T}\end{align*} \end{document} n , including counting the number of vertices on each face, and see [17] for enumeration of the vertices, edges, and cells in terms of formulas using Fibonacci numbers.…”

A survey of alternating permutations

“…Since the homogeneous part of ϕ has determinant ±1, it follows that ϕ is a volume-preserving bijection from P n onto C n , so vol(C n ) = vol(P n ) = E n /n!. [13] compute the f -vector (which gives the number of faces of each dimension) of a generalization of the polytopes P n . Since P n and C n are affinely equivalent, this computation also gives the f -vector of C n .…”

“…E 4 (q) = q 2 + q 3 + 2q 4 + q 5 E 5 (q) = q 2 + 2q 3 + 3q 4 + 4q 5 + 3q 6 + 2q 7 + q 8 E 6 (q) = q 3 + 2q 4 + 5q 5 + 7q 6 + 9q 7 + 10q 8 + 10q 9 + 8q 10 + 5q 11 + 2q 12 + q 13 .…”

“…4.3]. Chebikin and Ehrenborg[13] compute the f -vector (which gives the number of faces of each dimension) of a generalization of the polytopes P n . Since P n and C n are affinely equivalent, this computation also gives the f -vector of C n .The polytope C n has an interesting connection to tridiagonal matrices.…”