Permutations sortable by deques and by two stacks in parallel

Abstract

Recently Albert and Bousquet-M\'elou \cite{AB15} obtained the solution to the long-standing problem of the number of permutations sortable by two stacks in parallel (tsip). Their solution was expressed in terms of functional equations. We show 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. Subject to plausible, but unproved, conditions, the radius of convergence of both generating functions is the same. Numerical work confirms this conjecture to 10 significant digits. Further numerical work suggests that the coefficients of the deque generating function behave as $\kappa_d \cdot \mu^n \cdot n^{-3/2},$ where $\mu = 8.281402207\ldots,$ while the coefficients of the corresponding tsip generating function behave as $\kappa_p \cdot \mu^n \cdot n^{\gamma}$ with $\gamma \approx -2.473.$ The constants $\kappa_d$ and $\kappa_p$ are also estimated. {\em Inter alia,} we study the asymptotics of quarter-plane loops, starting and ending at the origin, with weight $a$ given to north-west and east-south turns. The critical point varies continuously with $a,$ while the corresponding exponent variation is found to be continuous and monotonic for $a > -1/2,$ but discontinuous at $a=-1/2.$