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 tau-reduction argument where tau is an element of S-m is given and all patterns in T are assumed to contain tau. For these situations, cell decompositions are introduced and studied. We describe an observation which allows to reduce the determination of the generating function for vertical bar S-n(T)vertical bar to the determination of a set of generating functions for simpler problems. The usefulness of this approach is demonstrated by several examples.