Stanley, Richard P. f-vectors and h-vectors of simplicial posets. (English) Zbl 0727.06009 J. Pure Appl. Algebra 71, No. 2-3, 319-331 (1991). 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. Reviewer: J.Neggers (Tuscaloosa) Cited in 4 ReviewsCited in 39 Documents MSC: 06A11 Algebraic aspects of posets 05E99 Algebraic combinatorics 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Keywords:Cohen-Macaulay ring; Gorenstein complex; simplicial poset; simplicial complexes; f-vectors; Stanley-number; Stanley-function; Cohen-Macaulay simplicial complexes; Cohen-Macaulay simplicial poset; Gorenstein posets; h-vector PDF BibTeX XML Cite \textit{R. P. Stanley}, J. Pure Appl. Algebra 71, No. 2--3, 319--331 (1991; Zbl 0727.06009) Full Text: DOI References: [1] Atiyah, M.F.; MacDonald, I.G., Introduction to commutative algebra, (1969), Addison-Wesley Reading, MA · Zbl 0175.03601 [2] Baclawski, K., Rings with lexicographic straightening law, Adv. in math., 39, 185-213, (1981) · Zbl 0466.13004 [3] Björner, A., Posets, regular CW complexes and Bruhat order, European J. combin., 5, 7-16, (1984) · Zbl 0538.06001 [4] Björner, A., Face numbers of complexes and polytopes, Proceedings intenational congress of mathematicians, 1408-1418, (1987), Berkeley, 1986 [5] Björner, A.; Frankl, P.; Stanley, R., The number of faces of balanced Cohen-Macaulay complexes and a generalized Macaulay theorem, Combinatorica, 7, 23-34, (1987) · Zbl 0651.05010 [6] Björner, A.; Garsia, A.; Stanley, R., An introduction to the theory of Cohen-Macaulay partially ordered sets, (), 583-615 [7] DeConcini, C.; Eisenbud, D.; Procesi, C., Hodge algebras, Astérisque, 91, 1-87, (1982) [8] Eisenbud, D., Introduction to algebras with straightening laws, (), 243-268 [9] Garsia, A.M.; Stanton, D., Group actions on Stanley-Reisner rings and invariants of permutation groups, Adv. in math., 51, 107-201, (1984) · Zbl 0561.06002 [10] Hibi, T., Distributive lattices, semigroup rings and algebras with straightening laws, (), 93-109 [11] Kaplansky, I., Commutative rings, (1970), Allyn and Bacon Boston, MA · Zbl 0203.34601 [12] Matsumura, H., Commutative ring theory, (1986), Cambridge Univ. Press Cambridge [13] Stanley, R., Cohen-Macaulay complexes, (), 51-62 [14] Stanley, R., Hilbert functions of graded algebras, Adv. in math., 28, 57-83, (1978) · Zbl 0384.13012 [15] Stanley, R., Combinatorics and commutative algebra, () · Zbl 0838.13008 [16] Stanley, R., Enumerative combinatorics, Vol. I, (1986), Wadsworth and Brooks/Cole Monterey, CA · Zbl 0608.05001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.