The total run length of a word

TL;DR

This paper investigates the total run length (TRL) in words, providing bounds for the maximum TRL, a formula for the average TRL over an alphabet, and exploring related combinatorial properties.

Contribution

It introduces the concept of total run length (TRL), establishes bounds for its maximum in words of length n, and derives formulas for average TRL over different alphabet sizes.

Findings

Maximum TRL bounds: n^2/8 < τ(n) < 47n^2/72 + 2n

Derived a formula for average TRL over an alphabet of size α

Provided new insights into the combinatorial structure of runs in words

Abstract

A run in a word is a periodic factor whose length is at least twice its period and which cannot be extended to the left or right (by a letter) to a factor with greater period. In recent years a great deal of work has been done on estimating the maximum number of runs that can occur in a word of length $n$. A number of associated problems have also been investigated. In this paper we consider a new variation on the theme. We say that the total run length (TRL) of a word is the sum of the lengths of the runs in the word and that $\tau(n)$ is the maximum TRL over all words of length $n$. We show that $n^2/8 < \tau(n) < 47n^2/72 + 2n$ for all $n$. We also give a formula for the average total run length of words of length $n$ over an alphabet of size $\alpha$, and some other results.