Sorting by shuffling methods and a queue
Stoyan Dimitrov

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 is the shuffle queue corresponding to the shuffling method , then we find a number of surprising results related to two natural variations of shuffle queues denoted by and . 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…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algorithms and Data Compression · Coding theory and cryptography
