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The set of f-vectors of pure simplicial complexes is an important but little understood object in combinatorics and combinatorial commutative algebra. Unfortunately, its explicit characterization appears to be a virtually intractable problem, and its structure is very irregular and complicated. The purpose of this note, where we combine a few different algebraic and combinatorial techniques, is to lend some further evidence to this fact. We first show that pure (in fact, Cohen-Macaulay) f-vectors can be nonunimodal with arbitrarily many peaks, thus improving the corresponding results known for level Hilbert functions and pure O-sequences. We provide both an algebraic and a combinatorial argument for this result. Then, answering negatively a question of the second author and collaborators posed in the recent AMS Memoir on pure O-sequences, we show that the interval property fails for the set of pure f-vectors, even in dimension 2. [ABSTRACT FROM AUTHOR]

Abstract: We conjecture that the set of all Hilbert functions of (artinian) level algebras enjoys a very natural form of regularity, which we call the Interval Conjecture (IC): If, for some positive integer α, and are both level h-vectors, then is also level for each integer . In the Gorenstein case, i.e. when , we also supply the Gorenstein Interval Conjecture (GIC), which naturally generalizes the IC, and basically states that the same property simultaneously holds for any two symmetric entries, say and , of a Gorenstein h-vector. These conjectures are inspired by the research performed in this area over the last few years. A series of recent results seems to indicate that it will be nearly impossible to characterize explicitly the sets of all Gorenstein or of all level Hilbert functions. Therefore, our conjectures would at least provide the existence of a very strong — and natural — form of order in the structure of such important and complicated sets. We are still far from proving the conjectures at this point. However, we will already solve a few interesting cases, especially when it comes to the IC, in this paper. Among them, that of Gorenstein h-vectors of socle degree 4, that of level h-vectors of socle degree 2, and that of non-unimodal level h-vectors of socle degree 3 and any given codimension. [Copyright &y& Elsevier]

The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen-Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture in the Gorenstein case. [ABSTRACT FROM AUTHOR]

We look at the number L (n) of O -sequences of length n. Recall that an O -sequence can be defined algebraically as the Hilbert function of a standard graded k -algebra, or combinatorially as the f -vector of a multicomplex. The sequence L (n) was first investigated in a recent paper by commutative algebraists Enkosky and Stone, inspired by Huneke. In this note, we significantly improve both of their upper and lower bounds, by means of a very short partition-theoretic argument. In particular, it turns out that, for suitable positive constants c 1 and c 2 and all n > 2 , e c 1 n ≤ L (n) ≤ e c 2 n log n. It remains an open problem to determine an exact asymptotic estimate for L (n). [ABSTRACT FROM AUTHOR]

Abstract: We determine new bounds on the entries of Gorenstein Hilbert functions, both in any fixed codimension and asymptotically. Our first main theorem is a lower bound for the degree entry of a Gorenstein h-vector, in terms of its entry in degree i. This result carries interesting applications concerning unimodality: indeed, an important consequence is that, given r and i, all Gorenstein h-vectors of codimension r and socle degree (this function being explicitly computed) are unimodal up to degree . This immediately gives a new proof of a theorem of Stanley that all Gorenstein h-vectors in codimension three are unimodal. Our second main theorem is an asymptotic formula for the least value that the ith entry of a Gorenstein h-vector may assume, in terms of codimension, r, and socle degree, e. This theorem broadly generalizes a recent result of ours, where we proved a conjecture of Stanley predicting that asymptotic value in the specific case and , as well as a result of Kleinschmidt which concerned the logarithmic asymptotic behavior in degree . [Copyright &y& Elsevier]

Abstract: We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function , we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from to , and Hesse configurations in . [Copyright &y& Elsevier]