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On the volume of a certain polytope

Clara S. Chan·David P. Robbins·David S. Yuen
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Abstract

Let n >= 2 be an integer and consider the set T_n of n by n permutation matrices pi for which pi_{ij}=0 for j>=i+2. In this paper we study the convex hull of T_n, which we denote by P_n. P_n is a polytope of dimension binom{n}{2}. Our main purpose is to provide evidence for the following conjecture concerning its volume. 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 for i varying from 0 to n-2 of frac{1}{i+1} binom{2i}{i}. That is, P_n is the product of v_n and the first n-1 Catalan numbers. We also give a related result on the Ehrhart polynomial of P_n.

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Clara S. Chan, David P. Robbins, David S. Yuen. 1998-10-29. On the volume of a certain polytope. https://arxiv.org/abs/math/9810154

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