Sorting by shuffling methods and a queue

TL;DR

This paper explores sorting permutations using shuffle queues based on various shuffling methods, revealing new equivalences, bounds, formulas, and conjectures related to permutation sorting capabilities of these devices.

Contribution

It introduces new models of shuffle queues, characterizes their sorting permutation sets, and provides formulas and bounds for their sorting capacities, including surprising connections to well-known permutation classes and Fibonacci numbers.

Findings

Sorting by a deque is equivalent to sorting by a shuffle queue that can reverse its content.

The set of permutations sortable by $ ext{cuts}^ ext{'}$ shuffle queues are the 321-avoiding separable permutations.

Number of permutations sortable by $ ext{pop}$-type shuffle queues relates to Fibonacci numbers.

Abstract

We study sorting by queues that can rearrange their content by applying permutations from a predefined set. These new sorting devices are called shuffle queues and we investigate those of them corresponding to sets of permutations defining some well-known shuffling methods. If $\mathbb{Q}_{\Sigma}$ is the shuffle queue corresponding to the shuffling method $\Sigma$, then we find a number of surprising results related to two natural variations of shuffle queues denoted by $\mathbb{Q}_{\Sigma}^{\prime}$ and $\mathbb{Q}_{\Sigma}^{\textsf{pop}}$. These require the entire content of the device to be unloaded after a permutation is applied or unloaded by each pop operation, respectively. First, we show that sorting by a deque is equivalent to sorting by a shuffle queue that can reverse its content. Next, we focus on sorting by cuts. We prove that the set of permutations that one can sort by using $\mathbb{Q}_{\text{cuts}}^{\prime}$ is the set of the $321$-avoiding separable permutations. We give lower and upper bounds to the maximum number of times the device must be used to sort a permutation. Furthermore, we give a formula for the number of $n$-permutations, $p_{n}(\mathbb{Q}_{\Sigma}^{\prime})$, that one can sort by using $\mathbb{Q}_{\Sigma}^{\prime}$, for any shuffling method $\Sigma$, corresponding to a set of irreducible permutations. We also show that $p_{n}(\mathbb{Q}_{\Sigma}^{\textsf{pop}})$ is given by the odd indexed Fibonacci numbers $F_{2n-1}$, for any shuffling method $\Sigma$ having a specific "back-front" property. The rest of the work is dedicated to a surprising conjecture inspired by Diaconis and Graham, which states that one can sort the same number of permutations of any given size by using the devices $\mathbb{Q}_{\text{In-sh}}^{\textsf{pop}}$ and $\mathbb{Q}_{\text{Monge}}^{\textsf{pop}}$, corresponding to the popular In-shuffle and Monge shuffling methods.