Generating Trees and the Catalan and Schr

A permutation Sn avoids the subpattern i has no subse quence having all the same pairwise comparisons as and we write Sn We present a new bijective proof of the well known result that jSn j jSn j cn the n th Catalan number A generalization to forbidden pat terns of length gives an asymptotic formula for the vexillary permutations We settle a conjecture of Shapiro and Getu that jSn j sn the Schr oder number and characterize the deque sortable permutations of Knuth also counted by sn Introduction to forbidden subsequences We regard a permutation Sn as a sequence of n elements f i g n i We say that contains the letter pattern i there is a triple i j k n such that k i j Otherwise avoids the pattern We de ne avoiding permutations similarly for every Sk De nition For Sk a permutation Sn is avoiding i there is no i i i k n such that i i ik The subsequence f i j g k j is said to have type Two sequences of length n are evidently of the same type i they have the same pairwise comparisons throughout namely if i j i j We denote by Sn the set of all permutations in Sn which avoid If R f qg we abbreviate Sn R Sn q T Sn j Fundamental questions are to determine jSn R j viewed as a function of n and if jSn R j jSn R j to discover an explicit bijection between Sn R and Sn R The most studied case has been to forbid a single pattern of length Because of obvious symmetry arguments described below there are only two distinct cases to enumerate jSn j and jSn j It happens that these two functions are equal jSn j jSn j cn n n n Historically these two enumerative results were obtained independently The rst satisfactory bijection between the two cases was presented by Rodica Simion and Frank Schmidt and a second was given by Dana Richards