Permutations sortable by deques and by two stacks in parallel

TL;DR

This paper relates the enumeration of permutations sortable by deques to those sortable by two stacks in parallel, using functional equations, and explores their asymptotic behaviors and generating functions.

Contribution

It establishes a connection between deque and two-stack in parallel permutation sorting problems via functional equations and analyzes their asymptotic properties.

Findings

Radius of convergence of generating functions is conjectured to be the same.

Coefficients for deque and tsip generating functions exhibit different asymptotic behaviors.

Numerical estimates of constants and exponents for asymptotic formulas are provided.

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 $a=-1/2.$