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.

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