Pure O-Sequences and Matroid h-Vectors |
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Authors: | Huy Tài Hà Erik Stokes Fabrizio Zanello |
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Institution: | 1. Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, LA, 70118, USA 2. Department of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, IN, 47803, USA 3. Department of Mathematics, MIT, Cambridge, MA, 02139-4307, USA 4. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931-1295, USA
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Abstract: | We study Stanley’s long-standing conjecture that the h-vectors of matroid simplicial complexes are pure O-sequences. Our method consists of a new and more abstract approach, which shifts the focus from working on constructing suitable artinian level monomial ideals, as often done in the past, to the study of properties of pure O-sequences. We propose a conjecture on pure O-sequences and settle it in small socle degrees. This allows us to prove Stanley’s conjecture for all matroids of rank 3. At the end of the paper, using our method, we discuss a first possible approach to Stanley’s conjecture in full generality. Our technical work on pure O-sequences also uses very recent results of the third author and collaborators. |
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