Generalized Stack Permutations

Stacks which allow elements to be pushed into any of the top r positions and popped from any of the top s positions are studied. An asymptotic formula for the number un of permutations of length n sortable by such a stack is found in the cases r=1 or s=1. This formula is found from the generating function of un. The sortable permutations are characterized if r=1 or s=1 or r=s=2 by a forbidden subsequence condition.