m at h . C O / 0 20 52 06 v 1 1 9 M ay 2 00 2 132 - avoiding Two - stack Sortable Permutations , Fibonacci Numbers , and Pell Numbers ∗

In [W2] West conjectured that there are 2(3n)!/((n+1)!(2n+1)!) two-stack sortable permutations on n letters. This conjecture was proved analytically by Zeilberger in [Z]. Later, Dulucq, Gire, and Guibert [DGG] gave a combinatorial proof of this conjecture. In the present paper we study generating functions for the number of two-stack sortable permutations on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary permutation τ on k letters. In several interesting cases this generating function can be expressed in terms of the generating function for the Fibonacci numbers or the generating function for the Pell numbers.