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Conference Papers Discrete Mathematics and Theoretical Computer Science Year : 2014

Sorting with two stacks in parallel

Abstract

At the end of the 1960s, Knuth characterised in terms of forbidden patterns the permutations that can be sorted using a stack. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Pratt and Tarjan asked about permutations that can be sorted using two stacks in parallel. This question is significantly harder, and the associated counting question has remained open for 40 years. We solve it by giving a pair of equations that characterise the generating function of such permutations. The first component of this system describes the generating function $Q(a,u)$ of square lattice loops confined to the positive quadrant, counted by the length and the number of North-West and East-South factors. Our analysis of the asymptotic number of sortable permutations relies at the moment on two intriguing conjectures dealing with this series. Given the recent activity on walks confined to cones, we believe them to be attractive $\textit{per se}$. We prove these conjectures for closed walks confined to the upper half plane, or not confined at all.
Nous énumérons les permutations triables par deux piles en parallèle. Cette question était restée ouverte depuis les travaux de Knuth, Pratt et Tarjan dans les années 70. Notre solution consiste en une paire d’équations qui caractérisent la série génératrice. La première composante de ce système décrit la série $Q(a,u)$ des chemins fermés confinés dans le quart de plan positif, comptés selon leur longueur et le nombre de facteurs Nord-Ouest ou Est-Sud. Notre analyse du comportement asymptotique du nombre de permutations triables repose à ce stade sur deux conjectures remarquables portant sur $Q(a; u)$. Nous les prouvons pour les chemins fermés non confinés, ou confinés au demi-plan supérieur.
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Dates and versions

hal-01207545 , version 1 (01-10-2015)

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Michael Albert, Mireille Bousquet-Mélou. Sorting with two stacks in parallel. 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014), 2014, Chicago, United States. pp.585-596, ⟨10.46298/dmtcs.2425⟩. ⟨hal-01207545⟩
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