Classifying Permutations under Context-Directed Swaps and the \textbf{cds} game

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 permutation provides, via the orbit-stabilizer theorem, enumerative results regarding the number of permutations with maximal strategic pile and a given number of \textbf{cds}-eligible contexts. Prior work introduced a natural two-person game on permutations that are not \textbf{cds}-sortable. The decision problem of which player has a winning strategy in a particular instance of the game appears to be of high computational complexity. Extending prior results, this work presents new conditions for player ONE to have a winning strategy in this combinatorial game.