The polytope of Tesler matrices

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.

26 pages, 4 figures. v3 has a corrected proof of Lemma 2.4, updated references

References in corpus (2)

• The Ring of Malcev-Neumann Series and the Residue Theorem
• A weighted sum over generalized Tesler matrices

Cited by in corpus (12)

• A combinatorial model for computing volumes of flow polytopes
• Flow-oriented perturbation theory
• Kostant's partition function and magic multiplex juggling sequences
• A Fuss-Catalan variation of the caracol flow polytope
• A weighted sum over generalized Tesler matrices
• Flow polytopes with Catalan volumes
• Volumes of flow polytopes related to caracol graphs
• Flow polytopes and the space of diagonal harmonics
• Refinements and Symmetries of the Morris identity for volumes of flow polytopes
• Ehrhart positivity of Tesler polytopes and Berline-Vergne's valuation
• Volumes of generalized Chan-Robbins-Yuen polytopes
• On the poset and asymptotics of Tesler Matrices