Sorting with two ordered stacks in series M. D. Atkinson a; M. M. Murphy b N. Ru skuc b Summary: Sorting with two ordered stacks in series M. D. Atkinson a; M. M. Murphy b N. Ruskuc b a Department of Computer Science, University of Otago, Dunedin, New Zealand b School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS Abstract The permutations that can be sorted by two stacks in series are considered, subject to the condition that each stack remains ordered. A forbidden characterisation of such permutations is obtained and the number of permutations of each length is determined by a generating function. Key words: Stacks, permutations, forbidden patterns, enumeration 1 Introduction The question of which permutations can be sorted by a single stack, and how many there are of each length, was solved by Knuth in [5]. He showed that a permutation is stack sortable if and only if it has no subpermutation 231 (i.e. subsequence order isomorphic to 231) and that the number of such permutations of length n is the n th Catalan number. At the same time he also introduced the problem of sorting permutations by two or more stacks in series and this was subsequently investigated further by Tarjan in [10]. Let S k denote the set of permutations that can be sorted by k stacks in series. Collections: Computer Technologies and Information Sciences