Enumerating permutations sortable by $k$ passes through a pop-stack

TL;DR

This paper proves that the generating functions for permutations sortable by k passes through a pop-stack are rational for any k, and provides an algorithm to compute these functions explicitly for small k.

Contribution

It establishes the rationality of the generating functions P_k(x) for all k and introduces an algorithm to derive these functions, extending previous results.

Findings

P_k(x) is rational for all k

Explicit formulas for P_k(x) for k ≤ 6

Algorithm to compute P_k(x) for any k

Abstract

In an exercise in the first volume of his famous series of books, Knuth considered sorting permutations by passing them through a stack. Many variations of this exercise have since been considered, including allowing multiple passes through the stack and using different data structures. We are concerned with a variation using pop-stacks that was introduced by Avis and Newborn in 1981. Let $P_k(x)$ be the generating function for the permutations sortable by $k$ passes through a pop-stack. The generating function $P_2(x)$ was recently given by Pudwell and Smith (the case $k=1$ being trivial). We show that $P_k(x)$ is rational for any $k$. Moreover, we give an algorithm to derive $P_k(x)$, and using it we determine the generating functions $P_k(x)$ for $k\leq 6$.