Permutation Patterns and Cell Decompositions

Let S_n be the symmetric group of all permutations of n letters, and let S_n(T) be the set of those permutations which avoid a given set of patterns T. In the present paper, we consider a τ-reduction argument where τ∈ S_m is given and all patterns in T are assumed to contain τ. For these situations, cell decompositions are introduced and studied. We describe an observation which allows to reduce the determination of the generating function for | S_n(T) | to the determination of a set of generating functions for simpler problems. The usefulness of this approach is demonstrated by several examples.