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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.

MSC:
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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[1] Armstrong, D.: Tesler matrices. Talk slides Bruce Saganfest. http://www.math.miami.edu/ armstrong/Talks/Tesler_Saganfest.pdf (2014)
[2] Armstrong, D; Garsia, A; Haglund, J; Rhoades, B; Sagan, B, Combinatorics of tesler matrices in the theory of parking functions and diagonal harmonics, J. Comb., 3, 451-494, (2012) · Zbl 1291.05203
[3] Baldoni-Silva, W., Beck, M., Cochet, C., Vergne, M.: Volume computation for polytopes and partition functions for classical root systems. Discrete Comput. Geom. 35, 551-595 (2006). Maple worksheets: http://www.math.jussieu.fr/ vergne/work/IntegralPoints.html · Zbl 1105.52001
[4] Baldoni, W; Vergne, M, Kostant partitions functions and flow polytopes, Transform. Groups, 13, 447-469, (2008) · Zbl 1200.52008
[5] Baldoni, W; Vergne, M, Morris identities and the total residue for a system of type \(A_r\), Prog. Math., 220, 1-19, (2004) · Zbl 1097.33011
[6] Brion, M; Vergne, M, Residue formulae, vector partition functions and lattice points in rational polytopes, J. Am. Math. Soc., 10, 797-833, (1997) · Zbl 0926.52016
[7] Butler, F., Can, M., Haglund, J., Remmel, J.: Rook Theory Notes. http://www.math.ucsd.edu/ remmel/files/Book.pdf · Zbl 1034.52011
[8] Carlsson, E., Mellit, A.: A proof the shuffle conjecture. arXiv:1508.06239 · Zbl 1387.05265
[9] Chan, CS; Robbins, DP, On the volume of the polytope of doubly stochastic matrices, Exp. Math., 8, 291-300, (1999) · Zbl 0951.05015
[10] Chan, CS; Robbins, DP; Yuen, DS, On the volume of a certain polytope, Exp. Math., 9, 91-99, (2000) · Zbl 0960.05004
[11] Garsia, AM; Haglund, J, A positivity result in the theory of Macdonald polynomials, Proc. Natl. Acad. Sci. USA, 98, 4313-4316, (2001) · Zbl 1066.05144
[12] Garsia, AM; Haglund, J, A proof of the \(q, t\)-Catalan positivity conjecture, Discrete Math., 256, 677-717, (2002) · Zbl 1028.05115
[13] Garsia, AM; Haglund, J; Xin, G, Constant term methods in the theory of tesler matrices and Macdonald polynomial operators, Ann. Comb., 18, 83-109, (2014) · Zbl 1297.05240
[14] Gorsky, E; Negut, A, Refined knot invariants and Hilbert schemes, J. Math. Pures Appl., 104, 403-435, (2015) · Zbl 1349.14012
[15] Haglund, J, A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants, Adv. Math., 227, 2092-2106, (2011) · Zbl 1258.13020
[16] Haglund, J.: The \(q,t\)-Catalan Numbers and the Space of Diagonal Harmonics. University Lecture Series, vol. 41. American Mathematical Society (2008) · Zbl 1142.05074
[17] Haglund, J; Haiman, M; Loehr, N; Remmel, J; Ulyanov, A, A combinatorial formula for the character of the diagonal coinvariants, Duke Math. J., 126, 195-232, (2005) · Zbl 1069.05077
[18] Haglund, J., Loehr, N.: A conjectured combinatorial formula for the Hilbert series for diagonal harmonics. Discrete Math. 298(1), 189-204 (2005) · Zbl 1070.05007
[19] Haglund J., Remmel J., Wilson A.T.: The delta conjecture. arXiv:1509.07058 · Zbl 1383.05308
[20] Hille, L, Quivers, cones and polytopes, Linear Algebra Appl., 365, 215-237, (2003) · Zbl 1034.52011
[21] Klee, V., Witzgall, C.: Facets and Vertices of Transportation Polytopes, Mathematics of the Decision Sciences Part 1. Lectures in Applied Mathematics, vol. 11, pp. 257-282. AMS, Providence, RI (1968) · Zbl 0184.44601
[22] Levande, P.: Special cases of the parking functions conjecture and upper-triangular matrices. In: DMTCS proceedings, 23rd international conference on formal power series and algebraic combinatorics (FPSAC 2011), pp.635-644 (2011) · Zbl 1355.05259
[23] Morris, W.G.: Constant term identities for finite and affine root systems: conjectures and theorems. Ph.D. thesis, University of Wisconsin-Madison (1982) · Zbl 1200.52008
[24] Pitman, J; Stanley, RP, A polytope related to empirical distributions, plane trees, parking functions, and the associahedron, Discrete Comput. Geom., 27, 603-634, (2002) · Zbl 1012.52019
[25] Sloane, Neil J.A.: The on-line encyclopedia of integer sequences. http://oeis.org/ · Zbl 1159.11327
[26] Wilson, A.T.: A weighted sum over generalized Tesler matrices. arXiv:1510.02684 · Zbl 1362.05135
[27] Xin, G.: The ring of Malcev-Neumann series and the residue theorem. Ph.D. thesis, Brandeis University (2004)
[28] Zeilberger, D, Proof of conjecture of chan, robbins, and yuen, Electron. Trans. Numer. Anal., 9, 147-148, (1999) · Zbl 0941.05006
[29] Ziegler, G.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, New York (1995) · Zbl 0823.52002
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