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The polytope of Tesler matricesAlgebra & Discrete Mathematics
Tesler matrices are upper triangular matrices with nonnegative integer entries with certain restrictions on the sums of their rows and columns. Glenn Tesler studied these matrices in the 1990s and in 2011 Jim Haglund rediscovered them in his study of diagonal harmonics. Then in 2013, Gorsky and Negut found a different connection between these matrices and diagonal harmonics. We investigate a polytope whose integer points are the Tesler matrices. It turns out that this polytope is a flow polytope of the complete graph thus relating its lattice points to vector partition functions. We study the face structure of this polytope and show that it is a simple polytope. We show its h-vector is given by Mahonian numbers and its volume is a product of consecutive Catalan numbers and the number of Young tableaux of staircase shape.
This is joint work with Brendon Rhoades and Karola Mészàros.