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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
##### Software:
CoCoA; Macaulay2; SageMath
Full Text:
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