Permutations sortable by two stacks in parallel and quarter plane walks

At the end of the 1960s, Knuth characterised the permutations that can be sorted using a stack in terms of forbidden patterns. He also showed that they are in bijection with Dyck paths and thus counted by the Catalan numbers. Subsequently, Even & Itai, Pratt and Tarjan studied permutations that can be sorted using two stacks in parallel. This problem is significantly harder. In particular, a sortable permutation can now be sorted by several distinct sequences of stack operations. Moreover, 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 functional equations that characterise 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)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)Q(a,u). We prove that they hold for loops confined to the upper half plane, or not confined at all. They remain open for quarter plane loops. Given the recent activity on walks confined to cones, we believe them to be attractive per se.