The polytope of Tesler matrices

TL;DR

This paper introduces the Tesler polytope, connects it to flow polytopes and Kostant partition functions, and explores its combinatorial structure, faces, and volume, revealing links to Mahonian numbers, Catalan numbers, and Young tableaux.

Contribution

It characterizes the Tesler polytope as a flow polytope, describes its faces with Tesler tableaux, and computes its volume and h-vector, establishing new combinatorial connections.

Findings

Number of Tesler matrices counted by Kostant partition function

h-vector of Tesler polytope given by Mahonian numbers

Volume of Tesler polytope expressed via Catalan numbers and Young tableaux

Abstract

We introduce the Tesler polytope Tes_n(a_1,a_2,...,a_n), whose integer points are the Tesler matrices of size n with nonnegative integer hook sums a_1,a_2,...,a_n. We show that Tes_n(a) is a flow polytope and therefore the number of Tesler matrices is counted by the type A_n Kostant partition function evaluated at (a_1,a_2,...,a_n,-a_1-...-a_n). We describe the faces of this polytope in terms of "Tesler tableaux" and characterize when the polytope is simple. We prove that the h-vector of Tes_n(a) when all a_i>0 is given by the Mahonian numbers and calculate the volume of Tes_n(1,1,...,1) to be a product of consecutive Catalan numbers multiplied by the number of standard Young tableaux of staircase shape.