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


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.


Download Citation

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


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

zbMATH: 0960.05004
MathSciNet: MR1758803

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

Rights: Copyright © 2000 A K Peters, Ltd.


Vol.9 • No. 1 • 2000
Back to Top