Enumeration and asymptotics of restricted compositions having the same number of parts

Cyril Banderier (1) , Pawel Hitczenko (2)

1 LIPN - Laboratoire d'Informatique de Paris-Nord
2 Departments of Mathematics and Computer Science

We study pairs and m-tuples of compositions of a positive integer n with parts restricted to a subset P of positive integers. We obtain some exact enumeration results for the number of tuples of such compositions having the same number of parts. Under the uniform probability model, we obtain the asymptotics for the probability that two or, more generally, m randomly and independently chosen compositions of n have the same number of parts. For a large class of compositions, we show how a nice interplay between complex analysis and probability theory allows to get full asymptotics for this probability. Our results extend an earlier work of Bóna and Knopfmacher. While we restrict our attention to compositions, our approach is also of interest for tuples of other combinatorial structures having the same number of parts.

Mots clés: Integer composition; Pairs of combinatorial structures; Local limit theorem; Asymptotics of D-finite sequences; Diagonal of algebraic generating function

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

Cyril Banderier, Pawel Hitczenko. Enumeration and asymptotics of restricted compositions having the same number of parts. Discrete Applied Mathematics, 2012, pp.1-13. ⟨10.1016/j.dam.2011.12.011⟩. ⟨hal-00664192⟩