Prefix frequency of lost positions

TL;DR

This paper studies the frequency of lost positions in word prefixes, developing an algorithm that, through extensive computer search, shows the asymptotic density of runs in binary words is less than approximately 0.9482.

Contribution

It introduces an algorithm to analyze lost positions in prefixes and provides a new upper bound on the density of runs in binary words.

Findings

Asymptotic density of runs in binary words is less than 183/193 (~0.9482).

Developed an algorithm for analyzing lost positions in word prefixes.

Extensive computer search supports the new density bound.

Abstract

The concept of "lost positions" is a recently introduced tool for counting the number of runs in words. We investigate the frequency of lost positions in prefixes of words. This leads to an algorithm that allows to show, using an extensive computer search, that the asymptotic density of runs in binary words is less than $183/193\approx 0.9482$.