## Mészáros, Karola and Morales, Alejandro H. and Rhoades, Brendon - The polytope of Tesler matrices

dmtcs:2475 - Discrete Mathematics & Theoretical Computer Science, January 1, 2015, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
The polytope of Tesler matrices

Authors: Mészáros, Karola and Morales, Alejandro H. and Rhoades, Brendon

We introduce the Tesler polytope $Tes_n(a)$, whose integer points are the Tesler matrices of size n with hook sums $a_1,a_2,...,a_n in Z_{\geq 0}$. 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 $(a_1,a_2,...,a_n,-\sum_{i=1}^n a_i)$. 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.

Volume: DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)
Section: Proceedings
Published on: January 1, 2015
Submitted on: November 21, 2016
Keywords: Tesler matrices,flow polytopes,Kostant partition function,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]