Permutations generated by a depth 2 and infinite stack in series are algebraic

TL;DR

This paper proves that permutations generated by a depth 2 and infinite stack in series correspond to an unambiguous context-free language, leading to an algebraic generating function and explicit enumeration formulas.

Contribution

It establishes a bijection between these permutations and a context-free language, enabling explicit enumeration and analysis of their growth rate.

Findings

The permutation class has an algebraic generating function.

The explicit generating function is derived and expressed in closed form.

The growth rate of the permutation class approaches 2+2√5.

Abstract

We prove that the class of permutations generated by passing an ordered sequence $12\dots n$ through a stack of depth 2 and an infinite stack in series is in bijection with an unambiguous context-free language, where a permutation of length $n$ is encoded by a string of length $3n$. It follows that the sequence counting the number of permutations of each length has an algebraic generating function. We use the explicit context-free language to compute the generating function: \begin{align*} \sum_{n\geq 0} c_n t^n &= \frac{(1+q)\left(1+5q-q^2-q^3-(1-q)\sqrt{(1-q^2)(1-4q-q^2)}\right)}{8q} \end{align*} where $c_n$ is the number of permutations of length $n$ that can be generated, and $q \equiv q(t) = \frac{1-2t-\sqrt{1-4t}}{2t}$ is a simple variant of the Catalan generating function. This in turn implies that $c_n^{1/n} \to 2+2\sqrt{5}$.