Chain decomposition and the flag \(f\)-vector. (Q1399906)

Ehrenborg introduced a quasi-symmetric function encoding, denoted \(F_P\), for the flag \(f\)-vector of any finite, graded poset \(P\) with elements \(\widehat{0}\) and \(\widehat{1}\) in \textit{R. Ehrenborg} [Adv. Math. 119, No. 1, 1--25 (1996; Zbl 0851.16033)]. R. P. Stanley showed that for locally rank-symmetric \(P\), \(F_P\) is symmetric and asked for conditions to insure Schur-positivity of \(F_P\) [Electron. J. Comb. 3, No. 2, Research paper R6 (1996); printed version J. Comb. 3, No. 2, 161--182 (1996; Zbl 0857.05091)]. This paper gives chain decompositions and consequent flag \(f\)-vector formulas in terms of symmetric functions for several classes of posets. For noncrossing partition lattices, classical reflection groups, and two generalizations of posets of shuffles, the obtained \(f\)-vector formulas exhibit Schur-positivity, the chain decompositions moreover yield symmetric chain decompositions, shellability and supersolvability results. Combinatorial applications are also given. For graded monoid posets it is shown that the \(f\)-vector may not be Schur-positive.