Idempotent n-permutable varieties

One of the important classes of varieties identified in tame congruence theory is the class of varieties which are n-permutable for some n. In this paper, we prove two results: ( 1) for every n > 1, there is a polynomial-time algorithm that, given a finite idempotent algebra A in a finite language, determines whether the variety generated by A is n-permutable and ( 2) a variety is n-permutable for some n if and only if it interprets an idempotent variety that is not interpretable in the variety of distributive lattices.