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We study pairs and m--tuples of compositions of a positive integer n withparts restricted to a subset P of positive integers. We obtain some exactenumeration results for the number of tuples of such compositions having thesame number of parts. Under the uniform probability model, we obtain theasymptotics for the probability that two or, more generally, m randomly andindependently chosen compositions of n have the same number of parts. For alarge class of compositions, we show how a nice interplay between complexanalysis and probability theory allows to get full asymptotics for thisprobability. Our results extend an earlier work of B\'ona and Knopfmacher.While we restrict our attention to compositions, our approach is also ofinterest for tuples of other combinatorial structures having the same number ofparts.

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