Chain decomposition and the flag f-vector

Ehrenborg introduced a quasi-symmetric function encoding, denoted FP, for the flag f-vector of any finite, graded poset P with 0̂ and 1̂. Stanley observed that FP is a symmetric function whenever P is locally rank-symmetric and asked for conditions under which FP is Schur-positive. We provide formulas for FP for three classes of locally rank-symmetric posets: graded monoid posets, generalized posets of shuffles and noncrossing partition lattices for classical reflection groups. Our flag f-vector expressions for generalized shuffle posets and noncrossing partition lattices exhibit Schur-positivity, while graded monoid posets do not always have Schur-positive flag f-vector. f our flag f-vector expressions results from a poset chain decomposition. For the noncrossing partition lattices and shuffle posets, these also yield symmetric chain decompositions (by restriction to 1-chains), shellability and supersolvability results and combinatorial formulae including characteristic polynomial and zeta polynomial. Our (more complicated) flag f-vector expression for graded monoid posets involves Gröbner bases and a weighted notion of Möbius function for the poset of partitions of a multiset and related multiset intersection posets.