Permutations Avoiding a Simsun Pattern

A permutation π avoids the simsun pattern τ if π avoids the consecutive pattern τ and the same condition applies to the restriction of π to any interval [k]. Permutations avoiding the simsun pattern 321 are the usual simsun permutation introduced by Simion and Sundaram. Deutsch and Elizalde enumerated the set of simsun permutations that avoid in addition any set of patterns of length 3 in the classical sense. In this paper we enumerate the set of permutations avoiding any other simsun pattern of length 3 together with any set of classical patterns of length 3. The main tool in the proofs is a massive use of a bijection between permutations and increasing binary trees.