Counting occurrences for a finite set of words: combinatorial methods.

Frédérique Bassino (1) , Julien Clément (2) , Julien Fayolle (1) , Pierre Nicodème (3)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 Equipe AMACC - Laboratoire GREYC - UMR6072
3 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

In this article, we give the multivariate generating function counting texts according to their length and to the number of occurrences of words from a finite set. The application of the inclusion- exclusion principle to word counting due to Goulden and Jackson (1979, 1983) is used to derive the result. Unlike some other techniques which suppose that the set of words is reduced (i.e., where no two words are factor of one another), the finite set can be chosen arbitrarily. Noonan and Zeilberger (1999) already provided a Maple package treating the non-reduced case, without giving an expression of the generating function or a detailed proof. We provide a complete proof validating the use of the inclusion-exclusion principle. We also restate in modern terms the normal limit laws theorems of Bender and Kochman (1993), emphasising on the underlying analytic mean shifting method.

Mots clés: Word Statistics; Inclusion-Exclusion; Generating Functions; Aho-Corasick Automaton

Domaines: Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]; Combinatoire [math.CO]; Probabilités [math.PR]

Frédérique Bassino, Julien Clément, Julien Fayolle, Pierre Nicodème. Counting occurrences for a finite set of words: combinatorial methods.. 2009. ⟨hal-00452694v1⟩