Longest run of equal parts in a random integer composition

This paper examines a problem in enumerative and asymptotic combinatorics involving the classical structure of integer compositions. What is sought is an analysis on average and in distribution of the length of the longest run of consecutive equal parts in a composition of size nn. The problem was posed by Herbert Wilf at the Analysis of Algorithms conference in July 2009 (see arXiv:0906.5196).