 
Summary: GENERALISED STACK PERMUTATIONS
M D Atkinson
School of Mathematical and Computational Sciences
North Haugh, St Andrews, Fife KY16 9SS
Abstract
Stacks which allow elements to be pushed into any of the top r posi
tions and popped from any of the top s positions are studied. An asymp
totic 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 char
acterised if r = 1 or s = 1 or r = s = 2 by a forbidden subsequence
condition.
1 Introduction
Let = [ 1 ; 2 ; : : : ; n ] be a permutation of 1; 2 : : : ; n appearing as the input
stream to a stack. If, through an appropriate series of push and pop operations,
the stack can discharge the input elements in the order 1; 2 : : : ; n then is
said to be a stack sortable permutation. Stack sortable permutations were rst
investigated by Knuth in [4], section 2.2.1, and it was proved that there are
2n
n
