On the volume of a certain polytope

Abstract

Let n ≥ 2 be an integer and consider the set Tn of n × n permutation matrices π for which πij = 0 for j ≥ i + 2. We study the convex hull Pn of Tn/a polytope of dimension (Formula Presented). We provide evidence for several conjectures involving Pn, including Conjecture 1: Let vn denote the minimum volume of a simplex with vertices in the affine lattice spanned by Tn. Then the volume of Pn is vn times the product (Formula Presented) of the first n – 1 Catalan numbers. We also give a related result on the Ehrhart polynomial of Pn. © A K Peters, Ltd.