Experimental Mathematics

On the volume of a certain polytope

Clara S. Chan, David P. Robbins, and David S. Yuen

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.

Article information

Source
Experiment. Math., Volume 9, Issue 1 (2000), 91-99.

Dates
First available in Project Euclid: 5 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1046889594

Mathematical Reviews number (MathSciNet)
MR1758803

Zentralblatt MATH identifier
0960.05004

Subjects
Primary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]
Secondary: 05A10: Factorials, binomial coefficients, combinatorial functions [See also 11B65, 33Cxx] 52A38: Length, area, volume [See also 26B15, 28A75, 49Q20] 52A40: Inequalities and extremum problems

Citation

Chan, Clara S.; Robbins, David P.; Yuen, David S. On the volume of a certain polytope. Experiment. Math. 9 (2000), no. 1, 91--99. https://projecteuclid.org/euclid.em/1046889594


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