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Generic and special constructions of pure \(O\)-sequences. (English) Zbl 1309.05188
The main motivation for this article is a conjecture by R. P. Stanley [in: Higher Comb., Proc. NATO Adv. Study Inst., Berlin (West) 1976, 51–62 (1977; Zbl 0376.55007)] that the \(h\)-vector of a matroid complex is a pure \(O\)-sequence. Given a matroid \(M = (E,\mathcal{I})\), then \(\mathcal{I}\) is a simplicial complex, called the matroid complex of \(M\), where for every subset \(F\subseteq E\) the induced simplicial complex \(\mathcal{I}_F\) is pure (all the maximal faces have the same dimension). A Hilbert function of an Artinian algebra \(k[x_1,\ldots,x_n]/I\), or an \(O\)-sequence, is pure if \(I\) is a monomial ideal in which all the maximal elements, w.r.t. divisibility, have the same degree.
In this article, it is shown that the \(h\)-vectors of Stanley-Reisner rings of three classes of matroid complexes are indeed pure \(O\)-sequences:
(i) matroid complexes that are truncations of other matroid complexes (by passing on to their skeleton of smaller dimension),
(ii) matroid complexes of rank \(k\) whose duals are \((k + 2)\)-partite, and lastly
(iii) matroid complexes of Cohen-Macaulay complexes of type at most 5. Here, the type of a matroid complex \(\mathcal{I}\) is the last entry of the \(h\)-vector of the dual matroid complex of \(\mathcal{I}\).
The authors discuss consequences of these results as a possible aid in a computational search for a counterexample to the mentioned motivating conjecture by Stanley. They have developed a small C++ library which can be used to enumerate pure \(O\)-sequences. They have also made extensive use of CoCoA, Macaulay 2, and Sage software packages. The authors state that they are not strong believers in the validity of Stanley’s conjecture.

MSC:
05E45 Combinatorial aspects of simplicial complexes
05E40 Combinatorial aspects of commutative algebra
05B35 Combinatorial aspects of matroids and geometric lattices
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13F55 Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes
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