zbMATH — the first resource for mathematics

On power ideals of transversal matroids and their “parking functions”. (English) Zbl 1419.05222
Summary: To a vector configuration one can associate a polynomial ideal generated by powers of linear forms, known as a power ideal, which exhibits many combinatorial features of the matroid underlying the configuration.
In this note we observe that certain power ideals associated to transversal matroids are, somewhat unexpectedly, monomial. Moreover, the (monomial) basis elements of the quotient ring defined by such a power ideal can be naturally identified with the lattice points of a remarkable convex polytope: a polymatroid, also known as generalized permutohedron. We dub the exponent vectors of these monomial basis elements “parking functions” of the corresponding transversal matroid.
We highlight the connection between our investigation and Stanley-Reisner theory, and relate our findings to Stanley’s conjectured necessary condition on matroid \(h\)-vectors.
05E40 Combinatorial aspects of commutative algebra
05E45 Combinatorial aspects of simplicial complexes
05B35 Combinatorial aspects of matroids and geometric lattices
52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.)
Full Text: DOI arXiv
[1] Adiprasito, K.; Huh, J.; Katz, E., Hodge theory for combinatorial geometries, Ann. of Math. (2), 188, 2, 381-452, (2018) · Zbl 1442.14194
[2] Ardila, F., Enumerative and algebraic aspects of matroids and hyperplane arrangements, (2003)
[3] Ardila, F.; Postnikov, A., Combinatorics and geometry of power ideals, Trans. Amer. Math. Soc., 362, 8, 4357-4384, (2010) · Zbl 1226.05019
[4] Atkin, A. O. L., Remark on a paper of Piff and Welsh, J. Comb. Theory, Ser. B, 13, 2, 179-182, (1972) · Zbl 0263.05031
[5] Baker, M.; Shokrieh, F., Chip-firing games, potential theory on graphs, and spanning trees, J. Comb. Theory, Ser. A, 120, 1, 164-182, (2013) · Zbl 1408.05089
[6] Berget, A., Products of linear forms and Tutte polynomials, European J. Combin., 31, 7, 1924-1935, (2010) · Zbl 1219.05032
[7] Björner, A., Matroid applications, 40, Homology and shellability of matroids and geometric lattices, 226-283, (1992), Cambridge Univ. Press, Cambridge · Zbl 0772.05027
[8] Brualdi, R. A., Combinatorial geometries, 29, Transversal matroids, 72-97, (1987), Cambridge Univ. Press, Cambridge · Zbl 0631.05014
[9] Constantinescu, A.; Kahle, T.; Varbaro, M., Generic and special constructions of pure \(O\)-sequences, Bull. Lond. Math. Soc., 46, 5, 924-942, (2014) · Zbl 1309.05188
[10] Dahmen, W.; Micchelli, C. A., On the local linear independence of translates of a box spline, Studia Math., 82, 3, 243-263, (1985) · Zbl 0545.41018
[11] De Boor, C.; Dyn, N.; Ron, A., On two polynomial spaces associated with a box spline, Pacific J. Math., 147, 2, 249-267, (1991) · Zbl 0678.41009
[12] De Concini, C.; Procesi, C., Hyperplane arrangements and box splines, Michigan Math. J., 57, 201-225, (2008) · Zbl 1181.41011
[13] De Concini, C.; Procesi, C., Topics in hyperplane arrangements, polytopes and box-splines, xx+384 p. pp., (2010), Springer: Springer, New York · Zbl 1217.14001
[14] Gawrilow, E.; Joswig, M., Polytopes – combinatorics and computation (Oberwolfach, 1997), 29, polymake: a framework for analyzing convex polytopes, 43-73, (2000), Birkhäuser: Birkhäuser, Basel · Zbl 0960.68182
[15] Grayson, D. R.; Stillman, M. E., Macaulay2, a software system for research in algebraic geometry
[16] Holtz, O.; Ron, A., Zonotopal algebra, Adv. Math., 227, 2, 847-894, (2011) · Zbl 1223.13010
[17] Huh, J., \(h\)-vectors of matroids and logarithmic concavity, Adv. Math., 270, 49-59, (2015) · Zbl 1304.05013
[18] Lovász, L.; Plummer, M. D., Matching theory, 121, xxvii+544 p. pp., (1986), North-Holland Publishing Co., Amsterdam · Zbl 0618.05001
[19] Merino, C., The chip firing game and matroid complexes, Discrete models: combinatorics, computation, and geometry (Paris, 2001), AA, 245-255, (2001), Maison Inform. Math. Discrèt. (MIMD), Paris · Zbl 0998.05010
[20] Mohammadi, F.; Shokrieh, F., Divisors on graphs, binomial and monomial ideals, and cellular resolutions, Math. Z., 283, 1-2, 59-102, (2016) · Zbl 1336.05060
[21] Oh, S., Generalized permutohedra, \(h\)-vectors of cotransversal matroids and pure O-sequences, Electron. J. Combin., 20, 3, 14 p. pp., (2013) · Zbl 1295.52017
[22] Oxley, J., Matroid theory, 21, xiv+684 p. pp., (2011), Oxford University Press: Oxford University Press, Oxford · Zbl 1254.05002
[23] Postnikov, A.; Shapiro, B., Trees, parking functions, syzygies, and deformations of monomial ideals, Trans. Amer. Math. Soc., 356, 8, 3109-3142, (2004) · Zbl 1043.05038
[24] Schrijver, A., Combinatorial optimization. Polyhedra and efficiency, 24, i-xxxiv and 649-1217, (2003), Springer-Verlag: Springer-Verlag, Berlin · Zbl 1041.90001
[25] Stanley, R. P., Combinatorics and commutative algebra, 41, x+164 p. pp., (1996), Birkhäuser Boston, Inc.: Birkhäuser Boston, Inc., Boston, MA · Zbl 0838.13008
[26] The Sage Developers, SageMath, the Sage Mathematics Software System (Version 8.0), (2017)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.