On the volume of a certain polytope

1998/10/27 by Clara S. Chan, Chan, Clara S., David P. Robbins +3 · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO

paper · pdf · doi:10.48550/arxiv.math/9810154

this version corrects a small number of typographical errors in the original

arxiv created 1998/10/29 · arxiv updated 2009/11/30

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 binomn2. 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 frac1i+1 binom2ii. 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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