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 /vbS_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 /vbS_n(3142,2413)/vb = s_{n -1}, the Schröder number, and characterize the deque-sortable permutations of Knuth, also counted by s_{n - 1}.