Permutations sortable by two stacks in parallel and quarter plane walks (conference version)

Michael Albert (1) , Mireille Bousquet-Mélou (2)

1 Department of Computer Science, University of Otago
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

At the end of the 1960s, Knuth characterized 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. A few years later, Pratt and Tarjan asked about permutations that can be sorted using two stacks in parallel. This question is significantly harder. In particular, a sortable permutation can now be sorted by several distinct sequences of stack operations. Moreover, Pratt proved that in order to be sortable, a permutation must avoid infinitely many patterns. The associated counting question has remained open for 40 years. We solve it by giving a pair of equations that characterizes the generating function of permutations that can be sorted with two parallel stacks. 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 the series $Q(a,u)$. We prove that these conjectures hold for closed walks confined to the upper half-plane, or not confined at all. They remain open for quarter plane walks. Given the recent activity on walks confined to cones, we believe them to be attractive per se.

Domaines: Combinatoire [math.CO]

Michael Albert, Mireille Bousquet-Mélou. Permutations sortable by two stacks in parallel and quarter plane walks (conference version). Formal power series and algebraic combinatorics 2014, Jun 2014, Chicago, United States. pp.51-62. ⟨hal-00962390⟩