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 $$W_t(n)$$Wt(n) and $$W_t(n,k)$$Wt(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 the growth rate of $$W_t(n)$$Wt(n) when $$t=3$$t=3 and when $$t=4$$t=4.