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

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 B'ona 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.

💡 Research Summary

The paper investigates the joint distribution of the number of parts in restricted integer compositions. A composition of a positive integer n is a sequence of positive integers whose sum equals n; the authors restrict the allowed part sizes to a prescribed subset P ⊆ℕ. For a fixed n they consider m independent random compositions drawn uniformly from all P‑restricted compositions of n and ask for the probability that all m compositions have exactly the same number of parts.

The authors begin by encoding P‑restricted compositions with the bivariate generating function

\