Abstract
In this paper we partition in classes the set of matroids of fixed dimension on a fixed vertex set. In each class we identify two special matroids, respectively with minimal and maximal h-vector in that class. Such extremal matroids also satisfy a long-standing conjecture of Stanley. As a byproduct of this theory we establish Stanley’s conjecture in various cases, for example the case of Cohen-Macaulay type less than or equal to 3.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Björner, A.: The homology and shellability of matroids and geometric lattices. In Matroid applications, volume 40 of Encyclopedia Mathematics and its Applications, Cambridge University Press, Cambridge, pp. 226–283 (1992)
Boij, M., Migliore, J., Mirò-Roig, R., Nagel, U., Zanello, F.: On the shape of a pure O-sequence. Mem. Am. Math. Soc. (to appear)
Bruns, W., Herzog, J.: Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics vol. 39. Cambridge University press, Cambridge (1993)
Chari, M.K.: Two decompositions in topological combinatorics with applications to matroid complexes. Trans. Am. Math. Soc. 349(10), 3925–3943 (1997)
CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, http://cocoa.dima.unige.it
Hà, H.T., Stokes, E., Zanello, F.: Pure O-sequences and matroid h-vectors Ann. Comb. 17(3), 495–508 (2013). http://arxiv.org/abs/1006.0325
Hartshorne, R.: A property of A-sequences. Bull. de laplace S.M.F. 94, 61–65 (1966)
Hibi, T.: What can be said about pure O-sequences?. J. Combin. Theory Ser. A 50(2), 319–322 (1989)
Hibi, T.: Face number inequalities for matroid complexes and Cohen-Macaulay types of Stanley-Reisner rings of distributive lattices. Pac. J. Math. 154(2), 253–264 (1992)
Iarrobino, A.: Compressed algebras: Artin algebras having given socle degrees and minimal length. Trans. Am. Math. Soc. 285, 337–378 (1984)
Massey, D., Simion, R., Stanley, R., Vertigan, D., Welsh, D., Ziegler, G.: Lê numbers of arrangements and matroid identities. J. Combin. Theory Ser. B 70(1), 118–133 (1997)
Merino, C.: The chip firing game and matroid complexes, Discrete Mathematics and Theoretical Computer Science. Procedings, AA, Maison de l’informatique et des mathématiques discrétes (MIMD), Paris, pp. 245–255 (2001)
Merino, C., Steven D. Noble, M. Ramirez-Ibanez, S.D., Villarroel, R.: On the structure of the h-vector of a paving matroid. European J. Combin. 33(8), 1787–1799 (2012). http://arxiv.org/abs/1008.2031
Minh, N.C., Trung, N.V.: Cohen-Macaulayness of monomial ideals and symbolic powers of Stanley-Reisner ideals. Adv. Math. 226, 1285–1306 (2011)
Oh, S.: Generalized permutohedra, h-vector of cotransversal matroids and pure O-sequences European J. Combin. 20(3), (2013). http://arxiv.org/abs/1005.5586
Oxley, J.: Matroid Theory, 2nd edn. Oxford Graduate Texts in Mathematics, vol. 21. Oxford University Press, Oxford (2011)
Schweig, J.: On the h-Vector of a Lattice Path Matroid. Electron. J. Combin. 17(1), (2010)
Speyer, D.: A matroid invariant via the K-theory of the Grassmannian. Adv. Math. 221(3), 882–913 (2009)
Stanley, R.P.: Cohen-Macaulay Complexes, in Higher Combinatorics, pp. 51–62. Reidel Dordrecht, Boston (1977)
Stanley, R.P.: Combinatorics and Commutative Algebra. Birkhäuser, Boston (1996)
Stokes, E.: The h-vector of matroids and the arithmetic degree of squarefree strongly stable ideals, Ph.D. thesis, University of Kentucky (2008)
Stokes, E.: The h-vectors of 1-dimensional Matroid Complexes and a Conjecture of Stanley. preprint (2009). http://arxiv.org/abs/0903.3569
Swartz, E.: Lower bounds for h-vectors of k-CM, independence, and broken circuit complexes. SIAM J. Discrete Math. 18(3), 647–661 (2004/2005)
Varbaro, M.: Symbolic powers and matroids. Proc. Am. Math. Soc. 139(7), 2357–2366 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Constantinescu, A., Varbaro, M. (2015). h-Vectors of Matroid Complexes. In: Benedetti, B., Delucchi, E., Moci, L. (eds) Combinatorial Methods in Topology and Algebra. Springer INdAM Series, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-20155-9_29
Download citation
DOI: https://doi.org/10.1007/978-3-319-20155-9_29
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-20154-2
Online ISBN: 978-3-319-20155-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)