Generating Functions for Permutations which Contain a Given Descent Set

A large number of generating functions for permutation statistics can be obtained by applying homomorphisms to simple symmetric function identities. In particular, a large number of generating functions involving the number of descents of a permutation σ, des(σ), arise in this way. For any given finite set S of positive integers, we develop a method to produce similar generating functions for the set of permutations of the symmetric group Sn whose descent set contains S. Our method will be to apply certain homomorphisms to symmetric function identities involving ribbon Schur functions.