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

We study pairs and mm-tuples of compositions of a positive integer nn with parts restricted to a subset PP 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, mm randomly and independently chosen compositions of nn 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.