Sorting Permutations: Games, Genomes, and Cycles

TL;DR

This paper explores specialized permutation sorting operations, their mathematical properties, and introduces combinatorial two-player games inspired by these operations, with applications in genome maintenance modeling.

Contribution

It generalizes previous characterizations of cds-sortability and introduces novel combinatorial games based on these sorting operations.

Findings

Generalized characterizations of cds-sortability.

Development of natural combinatorial two-player games.

Insights into molecular sorting events in ciliates.

Abstract

Permutation sorting, one of the fundamental steps in pre-processing data for the efficient application of other algorithms, has a long history in mathematical research literature and has numerous applications. Two special-purpose sorting operations are considered in this paper: context directed swap, abbreviated cds, and context directed reversal, abbreviated cdr. These are special cases of sorting operations that were studied in prior work on permutation sorting. Moreover, cds and cdr have been postulated to model molecular sorting events that occur in the genome maintenance program of certain species of single-celled organisms called ciliates. This paper investigates mathematical aspects of these two sorting operations. The main result of this paper is a generalization of previously discovered characterizations of cds-sortability of a permutation. The combinatorial structure underlying this generalization suggests natural combinatorial two-player games. These games are the main mathematical innovation of this paper.