Pure O-Sequences and Matroid h-Vectors

Abstract

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.

Generic and special constructions of pure O -sequences

“…Other than this our understanding is poor. Positive answers to Stanley's conjecture are known for short h-vectors [15,21], and for special classes of matroids [32,33,36]. In the present paper, we prove that Stanley's conjecture holds for matroids that are truncations of other matroids and for matroids whose h-vector (1, h 1 , .…”

Two unfortunate properties of pure $f$-vectors

“…Indeed, the IP was conjectured in [1] for pure O-sequences, and asked as a question (Question 9.4) for pure f -vectors (see also [17]). As for pure O-sequences, in [1] the IP was proved in degree 3, thus allowing a new approach to Stanley's matroid h-vector conjecture [7]. However, it was then disproved in large degree by A. Constantinescu and M. Varbaro [6].…”

“…The smallest counterexample to the IP given by Theorem 3.1 is when r = 7. Namely, (1, 7, 21, 7) is a pure f -vector (it corresponds to the Steiner triple system, or Fano plane, S(2, 3, 7)), and so are the Cohen-Macaulay f -vectors (1,7,12,7) and (1,7,13,7).…”

“…However, the f -vector (1,7,20,7) is not pure, thus violating the IP. In fact, it is a simple exercise to check that (1, 7, b, 7) is a pure f -vector if and only if b ∈ [12, 19] ∪ {21}.…”

Lexicographic shellability, matroids, and pure order ideals

“…Merino et al [15] proved the conjecture for paving matroids by noting that all but the last entry of the h-vector is determined by the dimension and the number of vertices of the matroid. Hà, Stokes, and Zanello [8] established the conjecture for matroids of rank three by studying properties of the level Artinian algebras. De Loera, Kemper, and Klee [7] proved the conjecture combinatorially for matroids of rank 3 and corank 2 by studying the lattice of flats.…”