Combinatorics of periods in strings

We consider the set Γn of all period sets of strings of length n over a finite alphabet. We show that there is redundancy in period sets and introduce the notion of an irreducible period set. We prove that Γn is a lattice under set inclusion and does not satisfy the Jordan–Dedekind condition. We propose the first efficient enumeration algorithm for Γn and improve upon the previously known asymptotic lower bounds on the cardinality of Γn. Finally, we provide a new recurrence to compute the number of strings sharing a given period set, and exhibit an algorithm to sample uniformly period sets through irreducible period set.