On the shape of a pure 𝑂-sequence

Pure O-Sequences and Matroid h-Vectors

“…In [2], the ICP was proved for socle degrees at most 3, which will be a crucial tool in our proof. We should also point out that, however, while the ICP remains open in most instances -for example, in three variables -it has recently been disproved in the four variable case by A. Constantinescu and M. Varbaro [7].…”

“…The following conjecture is referred to as the Interval Conjecture for Pure O-sequences (ICP), and was recently stated by the last author in collaboration with Boij, Migliore, Mirò-Roig and Nagel [2] (see [32] for the original formulation of the Interval Conjectures in the context of arbitrary level and Gorenstein algebras, where it is still wide open). In [2], the ICP was proved for socle degrees at most 3, which will be a crucial tool in our proof.…”

“…We may assume that ∆ is not a complete intersection, since Corollary 4.1 already took care of this case. We may also assume that ∆ is not a cone since Stanley's conjecture holds for 1-dimensional matroids (see, e.g., [2,26,27]). Moreover, the case init I ∆ ≥ 3 has been dealt with in Lemma 4.4.…”

“…Notice that part (b), if true, appears to be difficult to prove in arbitrary socle degree, given that very little is known on the "second half" of a pure O-sequence, and that unlike the first half, this can behave very pathologically (see [2]). As for part (a), it would also be of considerable independent interest to show this fact algebraically.…”

“…Over thirty years later, Stanley's conjecture is still wide open and mostly not understood, although a number of interesting partial results have been obtained; see [5,6,11,15,16,19,22,26,27,28,29]. As of today, the typical approach to Stanley's conjecture has been to, given the h-vector h of a matroid, explicitly construct a pure monomial order ideal (or equivalently, an artinian level monomial algebra; see [2]) with h-vector h. Our goal, assisted by recent progress on pure O-sequences and especially the Interval Property in socle degree 3 (see [2]), is to avoid the above constructions and instead begin the study of Stanley's conjecture under a new and more abstract perspective.…”