  We investigate the state complexity of the permutation operation, or the
commutative closure, on Alphabetical Pattern Constraints (APC). This class
corresponds to level $3/2$ of the Straubing-Th{\'e}rien Hierarchy and includes
the finite, the piecewise-testable, or $\mathcal J$-trivial, and the $\mathcal
R$-trivial and $\mathcal L$-trivial languages. We give a sharp state complexity
bound expressed in terms of the longest strings in the unary projection
languages of an associated finite language and which is already sharp for the
subclass of finite languages. Additionally, for two subclasses, we give sharp
bounds expressed in terms of the size of a recognizing input automaton and the
size of the alphabet. Lastly, we investigate the inclusion and universality
problem on APCs up to permutational equivalence, two problems known to be
PSPACE-complete on APCs even for fixed alphabets in general, and show them to
be decidable in polynomial time for fixed alphabets in this case.
