Face numbers of generalized balanced Cohen-Macaulay complexes (Q2428635)

A basic invariant of a simplicial complex \(\Delta\) is its \(f\)-vector, \(f(\Delta) = (f_{-1},f_0, \ldots ,f_{\dim \Delta})\), where \(f_i\) denotes the number of \(i\)-dimensional faces of \(\Delta\). \textit{J.~B.~Kruskal} [Math. Optimization Tech. 251--278 (1963; Zbl 0116.35102)] and \textit{G.~Katona} [Theory of Graphs, Proc. Colloquium Tihany, Hungary 1966, 187--207 (1968; Zbl 0313.05003)] characterized the set of all \( f\)-vectors for the family of all simplicial complexes and \textit{R.~P.~Stanley} [Higher Comb., Proc. NATO Adv. Study Inst., Berlin (West) 1976, 51--62 (1977; Zbl 0376.55007)] the \(f\)-vectors of all Cohen-Macaulay (CM, for short) simplicial complexes. The main result of this paper is a common generalization of two theorems on the face numbers of CM simplicial complexes: the first is the theorem of \textit{R.~P.~Stanley} [Trans. Am. Math. Soc. 249, 139--157 (1979; Zbl 0411.05012)] (necessity) and \textit{A.~Björner}, \textit{P.~Frankl} and \textit{R.~Stanley} [Combinatorica 7, 23--34 (1987; Zbl 0651.05010)] (sufficiency) that characterizes all possible face numbers of \textbf{a}-balanced CM complexes, while the second is the theorem of \textit{I.~Novik} [Adv. Math. 192, No. 1, 183--208 (2005; Zbl 1092.52009)] (necessity) and \textit{J.~Browder} [J. Algebr. Comb. 32, No. 1, 99--112 (2010; Zbl 1228.05324)] (sufficiency) that characterizes the face numbers of CM subcomplexes of the join of the boundaries of simplices.