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The polytope of Tesler matrices. (English) Zbl 1355.05271
Summary: We introduce the Tesler polytope \(\mathrm{Tes}_n(\mathbf{a})\), whose integer points are the Tesler matrices of size \(n\) with hook sums \(a_1,a_2,\dotsc,a_n \in \mathbb {Z}_{\geq 0}\). We show that \(\operatorname{Tes}_n(a)\) is a flow polytope and therefore the number of Tesler matrices is counted by the type \(A_n\) Kostant partition function evaluated at \((a_1,a_2,\dotsc,a_n,-\sum _{i=1}^n a_i)\). 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 \(\operatorname{Tes}_n(a)\) when all \(a_i>0\) is given by the Mahonian numbers and calculate the volume of \(\operatorname{Tes}_n(1,1,\dotsc,1)\) to be a product of consecutive Catalan numbers multiplied by the number of standard Young tableaux of staircase shape.

05E10 Combinatorial aspects of representation theory
05A05 Permutations, words, matrices
05A10 Factorials, binomial coefficients, combinatorial functions
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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