Classifying Permutations under Context-Directed Swaps and the \textbf{cds} game
G. Brown, A. Mitchell, R. Raghavan, J. Rogge, M. Scheepers

TL;DR
This paper classifies permutations based on their extbf{cds}-eligible contexts, analyzes their strategic pile, and explores a related combinatorial game, providing new characterizations, enumerations, and conditions for winning strategies.
Contribution
It introduces a detailed classification of permutations by extbf{cds}-eligible contexts, characterizes those with maximal strategic pile, and extends analysis of the extbf{cds} game with new winning strategy conditions.
Findings
Complete characterization of permutations with maximal strategic pile.
Enumeration results for permutations with specific extbf{cds}-eligible contexts.
New conditions for player ONE to have a winning strategy.
Abstract
A special sorting operation called Context Directed Swap, and denoted \textbf{cds}, performs certain types of block interchanges on permutations. When a permutation is sortable by \textbf{cds}, then \textbf{cds} sorts it using the fewest possible block interchanges of any kind. This work introduces a classification of permutations based on their number of \textbf{cds}-eligible contexts. In prior work an object called the strategic pile of a permutation was discovered and shown to provide an efficient measure of the non-\textbf{cds}-sortability of a permutation. Focusing on the classification of permutations with maximal strategic pile, a complete characterization is given when the number of \textbf{cds}-eligible contexts is close to maximal as well as when the number of eligible contexts is minimal. A group action that preserves the number of \textbf{cds}-eligible contexts of a…
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Taxonomy
TopicsGenome Rearrangement Algorithms · Algorithms and Data Compression · DNA and Biological Computing
