ON THE VOLUME OF A CERTAIN POLYTOPE

CLARA S. CHAN, DAVID P. ROBBINS, AND DAVID S. YUEN

Abstract. Let n ≥ 2 be an integer and consider the set T_{n} of n × n permutation matrices π for which π_{ij} = 0 for j ≥ i + 2.

We study the convex hull P_{n} of T_{n}, a polytope of dimension \binom{n}{2}. We provide evidence for several conjectures involving P_{n}, including Conjecture 1: Let v_{n} denote the minimum volume of a simplex with vertices in the affine lattice spanned by T_{n}. Then the volume of P_{n} is v_{n} times the product

∏_{i=0}^{n−2} \frac{1}{i + 1}({2i \atop i})

of the first n − 1 Catalan numbers.

We also give a related result on the Ehrhart polynomial of P_{n}.