On the volume of a certain polytope
Combinatorics
2007-05-23 v2
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.
Cite
@article{arxiv.math/9810154,
title = {On the volume of a certain polytope},
author = {Clara S. Chan and David P. Robbins and David S. Yuen},
journal= {arXiv preprint arXiv:math/9810154},
year = {2007}
}
Comments
this version corrects a small number of typographical errors in the original