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

TL;DR

This paper investigates the enumeration and asymptotic probability that multiple compositions of an integer share the same number of parts, using complex analysis and probability theory, extending previous work on compositions.

Contribution

It provides exact enumeration formulas and full asymptotics for the probability that multiple compositions have equal parts, generalizing earlier results.

Findings

Derived exact enumeration results for tuples of compositions with equal parts

Established asymptotic probabilities for multiple compositions sharing the same number of parts

Extended previous work to broader classes of compositions and combinatorial structures

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 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.