Combinatorics of Periods in Strings

: We consider the set (n) of all period sets of strings of length n over a nite alphabet. We show that there is redundancy in period sets and introduce the notion of an irreducible period set. We prove that (n) is a lattice under set inclusion and does not satisfy the Jordan-Dedekind condition. We propose the rst enumeration algorithm for (n) and improve upon the previously known asymptotic lower bounds on the cardinality of (n). Finally, we provide a new recurrence to compute the number of strings sharing a given period set. Mots cls : priode, chane de caractres, autocorrlation, combinatoire, treillis, numration. Keywords: period, string, autocorrelation, combinatorics, lattice, enumeration. Combinatorics of Periods in Strings Eric Rivals 1 and Sven Rahmann 2 1 L.I.R.M.M., CNRS U.M.R. 5506 161 rue Ada, F-34392 Montpellier Cedex 5, France, [email protected] 2 Max-Planck-Institut fr Molekulare Genetik, Dept. of Computational Molecular Biology, Ihnestrae 73, D-14195 Berlin, Germany [email protected] Abstract. We consider the set (n) of all period sets of strings of length n over a nite alphabet. We show that there is redundancy in period sets and introduce the notion of an irreducible period set. We prove that (n) is a lattice under set inclusion and does not satisfy the JordanDedekind condition. We propose the rst enumeration algorithm for (n) and improve upon the previously known asymptotic lower bounds on the cardinality of (n). Finally, we provide a new recurrence to compute the number of strings sharing a given period set.

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