GENERALISED STACK PERMUTATIONS M D Atkinson 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 Collections: Computer Technologies and Information Sciences