  The special purpose sorting operation, context directed swap (CDS), is an
example of the block interchange sorting operation studied in prior work on
permutation sorting. CDS has been postulated to model certain molecular sorting
events that occur in the genome maintenance program of some species of
ciliates. We investigate the mathematical structure of permutations not
sortable by the CDS sorting operation. In particular, we present substantial
progress towards quantifying permutations with a given strategic pile size,
which can be understood as a measure of CDS non-sortability. Our main results
include formulas for the number of permutations in $\textsf{S}_n$ with maximum
size strategic pile. More generally, we derive a formula for the number of
permutations in $\textsf{S}_n$ with strategic pile size $k$, in addition to an
algorithm for computing certain coefficients of this formula, which we call
merge numbers.
