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f-vectors and h-vectors of simplicial posets. (English) Zbl 0727.06009
A simplicial poset is a finite poset P with $${\hat 0}$$ such that every interval $$[{\hat 0},x]$$ is a boolean algebra. Simplicial posets are generalizations of simplicial complexes. Like degree sequences for graphs f-vectors are associated with simplicial posets. Thus, if P is a simplicial poset of rank d, then $$f(P)=(f_ 0,f_ 1,...,f_{d- 1})=| \{x\in P|$$ $$[{\hat 0},x]\cong B_{i+1}\}|$$, where $$B_{i+1}$$ is a boolean algebra of rank $$i+1$$. The author shows that (Theorem 2.1) if $$f=(f_ 0,...,f_{d-1})\in {\mathbb{Z}}^ d$$, then $$f=f(P)$$ for a simplicial poset iff $$f_ i\geq \binom{d}{i+1}$$ for $$1\leq i\leq d-1$$. It seems that $$| \{P|$$ $$f=f(P)\}| =S(f)$$ (say, the Stanley-number of f), where P is a simplicial poset and where $$P=Q$$ iff P and Q are isomorphic, is in fact an interesting number deserving further discussion. Since with $$f=(f_ 0,...,f_{d-1})$$ one may associate a monomial $$M(f)=x_ 0^{f_ 0}...x_{d-1}^{f_{d-1}}$$, there is an associated generating function $${\mathcal S}(x_ 0,x_ 1,...)=\sum_{f}S(f)M(f)$$ (say, the Stanley-function), whose properties should be interesting and which might be analysable using the methods for which the author of this paper is well-known indeed. The h-vector is defined from the f-vector by $$\sum^{d}_{i=0}f_{i-1}(x-1)^{d- i}=\sum^{d}_{i=0}h_ ix^{d-i}.$$ Then h-vectors have been characterized for Cohen-Macaulay simplicial complexes. In (Theorem 3.10) it is shown that if $$h=(h_ 0,...,h_ d)\in {\mathbb{Z}}^{d+1}$$ then $$h=h(P)$$ for a Cohen-Macaulay simplicial poset iff $$h_ 0=1$$ and $$h_ i\geq 0$$ for all i. For Gorenstein posets and the derived notion Gorenstein* poset sufficient conditions on the h-vector are also provided (Theorem 4.3), with further conditions conditioning necessity. The methods of proof are (co)homological-algebraic-combinatorial in nature in the clear style for which the author is well-known.

##### MSC:
 06A11 Algebraic aspects of posets 05E99 Algebraic combinatorics 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry)
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