State Complexity of Permutation and Related Decision Problems on Alphabetical Pattern Constraints

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.