Résumé
-We compare three natural generalizations of periodicity for degenerate strings, which we refer to as weak, medium and strong periodicity. We define the concept of total autocorrelations, which are quaternary vectors indicating these three notions of periodicity.
-We characterize the three families of period sets, as well as the family of total autocorrelations, for each alphabet size. In particular, we prove necessary conditions period sets should satisfy and, to prove sufficiency, we show how to construct a degenerate string which gives rise to particular period sets.
-For each notion of periodicity, we (asymptotically) count the number of period sets, by combining known techniques from partial words with recent results from number theory.
-Moreover, we show that all families of period sets, as well as the family of total autocorrelations, form lattices under a suitably defined partial ordering.
-We compute the population of weak, medium and strong period sets (i.e., the number of strings with that period set). We also compute the population of total autocorrelations.