  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.
