Longest Run of Equal Parts in a Random Integer Composition

TL;DR

This paper analyzes the average and distribution of the longest run of consecutive equal parts in integer compositions, addressing a problem posed by Herbert Wilf in combinatorics.

Contribution

It provides an asymptotic and probabilistic analysis of the longest run of equal parts in integer compositions, a problem recently posed by Wilf.

Findings

Derived the asymptotic distribution of the longest run

Established average length of the longest run in compositions

Connected combinatorial analysis to probabilistic models

Abstract

This note 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 n. The problem was recently posed by Herbert Wilf (see arXiv: 0906.5196).