132-avoiding Two-stack Sortable Permutations, Fibonacci Numbers, and Pell Numbers

In 1990 West conjectured that there are $2(3n)!\mathrm{/}((n+1)!(2n+1)!)$ two-stack sortable permutations on $n$ letters. This conjecture was proved analytically by Zeilberger in 1992. Later, Dulucq, Gire, and Guibert 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 $\tau$ 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.