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The \(f\)-vector of a representable-matroid complex is log-concave. (English) Zbl 1301.05382
Summary: We show that the \(f\)-vector of the matroid complex of a representable matroid is log-concave. This proves the representable case of a conjecture made by J. H. Mason [“Matroids: unimodal conjectures and Motzkin’s theorem”, in: D. J. A. Welsh (ed.) et al., Combinatorics. Proceedings of the conference on combinatorial mathematics held at the Mathematical Institute, Oxford, 1972. Southend-on-Sea: The Institute of Mathematics and its Applications. 207–220 (1972; Zbl 0469.05001)].

05E45 Combinatorial aspects of simplicial complexes
05C31 Graph polynomials
05B35 Combinatorial aspects of matroids and geometric lattices
Full Text: DOI
[1] Brylawski, Thomas, The Tutte polynomial. I. general theory, (Matroid Theory and Its Applications, (1982), Liguori Naples), 125-275 · Zbl 1302.05023
[2] Brylawski, Thomas; Oxley, James, The Tutte polynomial and its applications, (Matroid Applications, Encyclopedia Math. Appl., vol. 40, (1992), Cambridge Univ. Press Cambridge), 123-225 · Zbl 0769.05026
[3] Brylawski, Tom, The broken-circuit complex, Trans. Amer. Math. Soc., 234, 417-433, (1977) · Zbl 0368.05022
[4] Huh, June, Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs, J. Amer. Math. Soc., 25, 907-927, (2012) · Zbl 1243.14005
[5] Huh, June; Katz, Eric, Log-concavity of characteristic polynomials and the Bergman Fan of matroids, Math. Ann., 354, 1103-1116, (2012) · Zbl 1258.05021
[6] Lenz, Matthias, Matroids and log-concavity, (2013) · Zbl 1301.05382
[7] Mason, John H., Matroids: unimodal conjectures and motzkin╩╝s theorem, (Proc. Conf. Combinatorial Math., Math. Inst., Oxford, 1972 (Southend-on-Sea), (1972)), 207-220
[8] Oxley, James G., Matroid theory, Oxford Sci. Publ., (1992), Clarendon Press, Oxford University Press New York · Zbl 0784.05002
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