Beyond the Runs Theorem

TL;DR

This paper improves the upper bound on the number of runs in a binary word of length n, reducing it from n-3 to approximately 0.957n, using a new proof technique and computer search.

Contribution

It introduces a new proof technique combined with computer search to establish a tighter upper bound on runs in binary words.

Findings

Number of runs in a binary word of length n is at most (22/23)*n

The bound is tighter than the previous n-3 limit

Uses a novel combination of proof technique and computational search

Abstract

Recently, a short and elegant proof was presented showing that a binary word of length $n$ contains at most $n-3$ runs. Here we show, using the same technique and a computer search, that the number of runs in a binary word of length $n$ is at most $\frac{22}{23}n<0.957n$.