Permutations sortable by deques and two stacks in parallel share the same growth rate

Abstract

Recently Albert and Bousquet-M\'elou obtained the solution to the long-standing problem of the enumeration of permutations sortable by two stacks in parallel (2sip). Their solution was expressed in terms of functional equations. E.P. and Guttmann then showed that the equally long-standing problem of the number of permutations sortable by a double ended queue (deque) can be simply related to the solution of the same functional equations. They then conjectured that the radius of convergence of both generating functions is the same, and reduced this conjecture to a series of conjectures of Albert and Bousquet-M\'elou regarding a generating function for quarter-plane loops. In this note we prove that the two growth rates are equal, using a combinatorial argument on certain lattice paths which are in bijection with the two classes. As a corollary we prove that the generating function P(t) for permutations sortable by two stacks in parallel satisfies an inequality which was conjectured by Albert and Bousquet-M\'elou.