Generating trees and the Catalan and Schröder numbers

Abstract

A permutation π ϵ S n avoids the subpattern τ iff π has no subsequence having all the same pairwise comparisons as τ, and we write π ϵ S n(τ). We present a new bijective proof of the well-known result that /vb S n (123)/ vb = / vbS n (132)/ vb = c n , the nth Catalan number. A generalization to forbidden patterns of length 4 gives an asymptotic formula for the vexillary permutations. We settle a conjecture of Shapiro and Getu that /vb S n (3142,2413)/ vb = s n -1 , the Schröder number, and characterize the deque-sortable permutations of Knuth, also counted by s n - 1 .