On Simsun and Double Simsun Permutations Avoiding a Pattern of Length Three

A permutation σ∈σ_n is simsun if for all k, the subword of . restricted to {1, . . . , k} does not have three consecutive decreasing elements. The permutation . is double simsun if both σ and σ^-1 are simsun. In this paper, we present a new bijection between simsun permutations and increasing 1-2 trees, and show a number of interesting consequences of this bijection in the enumeration of pattern-avoiding simsun and double simsun permutations. We also enumerate the double simsun permutations that avoid each pattern of length three.