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

Authors:
Cyril Banderier;Paweł Hitczenko
Affiliations:
Laboratoire d'Informatique de Paris Nord, UMR CNRS 7030, Institut Galilée, Université Paris 13, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France;Department of Mathematics, Drexel University, Philadelphia, PA 19104, USA and Department of Computer Science, Drexel University, Philadelphia, PA 19104, USA
Venue:
Discrete Applied Mathematics
Year:
2012

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Abstract

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 Bona 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.