Restricted simsun permutations

A permutation is simsun if for all k, the subword of the one-line notation consisting of the k smallest entries does not have three consecutive decreasing elements. Simsun permutations were introduced by Simion and Sundaram, who showed that they are counted by the Euler numbers. In this paper we enumerate simsun permutations avoiding a pattern or a set of patterns of length 3. The results involve Motkzin, Fibonacci, and secondary structure numbers. The techniques in the proofs include generating functions, bijections into lattice paths and generating trees.