Preimages under the Stack-Sorting Algorithm

Abstract

We use a method for determining the number of preimages of any permutation under the stack-sorting map in order to obtain recursive upper bounds for the numbers $W_t(n)$ and $W_t(n,k)$ of $t$-stack sortable permutations of length $n$ and $t$-stack sortable permutations of length $n$ with exactly $k$ descents. From these bounds, we are able to significantly improve the best known upper bounds for $\underset{n\to \infty }{\lim }\sqrt[n]{W_t(n)}$ when $t=3$ and $t=4$.

15 pages, 3 figures