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2000 On the volume of a certain polytope
Clara S. Chan, David P. Robbins, David S. Yuen
Experiment. Math. 9(1): 91-99 (2000).

Abstract

Let $n \ge 2$ be an integer and consider the set $T_n$ of $n \times n$ permutation matrices $\pi$ for which $\pi_{ij}=0$ for $j\ge i+2$.

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 $$\prod_{i=0}^{n-2} \frac{1}{i+1}\BINOM{2i}{i} $$ of the first $n-1$ Catalan numbers.

We also give a related result on the Ehrhart polynomial of $P_n$.

Editor's note: After this paper was circulated, Doron Zeilberger proved Conjecture 1, using the authors' reduction of the original problem to a conjectural combinatorial identity, and sketched the proofs of two others. The problems and methodology presented here gain even further interest thereby.

Citation

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Clara S. Chan. David P. Robbins. David S. Yuen. "On the volume of a certain polytope." Experiment. Math. 9 (1) 91 - 99, 2000.

Information

Published: 2000
First available in Project Euclid: 5 March 2003

zbMATH: 0960.05004
MathSciNet: MR1758803

Subjects:
Primary: 52B20
Secondary: 05A10, 52A38, 52A40

Rights: Copyright © 2000 A K Peters, Ltd.

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