Branching flags, branching nets, and reverse matchings

AbstractThe relation among three classes of combinatorial objects parametrized by partitions is discussed: the branching flag is a tree-type generalization of the flag in a vector space over a finite field; the branching nest is its finite set-theoretical counterpart; and the reverse matching, equinumerous to branching nests, is a dual concept of the complete matching. A mapping will be constructed of the set of branching flags into that of branching nests, which will give a decomposition of the variety of branching flags into cells parametrized by reverse matchings. Its Poincaré polynomial is related to a refinement of q-Stirling numbers