Ranks for families of permutation theories

The notion of rank for families of theories, similar to Morley rank for fixed theories, serves as a measure of complexity for given families. There arises a natural problem of describing a rank hierarchy for a series of families of theories.
In this article, we answer the question posed and describe the ranks and degrees for families of theories of permutations with different numbers of cycles of a certain length. A number examples of families of permutation theories that have a finite rank are given, and it is constructed a family of permutation theories having a specified countable rank and degree $n$. It is proved that in the family of permutation theories any theory equals a theory of a finite structure or it is approximated by finite structures, i.e. any permutation theory on an infinite set is pseudofinite. Topological properties of the families under consideration were studied.