The distribution of longest run lengths in integer compositions

TL;DR

This paper derives a generating function for counting integer compositions with constraints on run lengths, extending combinatorial methods to analyze run distributions in compositions.

Contribution

It introduces a novel generating function for compositions with bounded run lengths, utilizing advanced subword avoidance techniques.

Findings

Provides explicit generating functions for compositions with run length restrictions

Extends combinatorial methods to analyze run distributions in compositions

Offers tools for counting compositions with specific run length constraints

Abstract

We find the generating function for $C(n,k,r)$, the number of compositions of $n$ into $k$ positive parts all of whose runs (contiguous blocks of constant parts) have lengths less than $r$, using recent generalizations of the method of Guibas and Odlyzko for finding the number of words that avoid a given list of subwords.