Runs of Ones in Binary Strings

TL;DR

This paper investigates the total counts of runs of specific lengths in binary strings and explores their relationship with compositions of integers, providing multiple methods for these calculations.

Contribution

It introduces three different computational approaches for counting runs of a given length in binary strings and connects this problem to integer compositions.

Findings

Derived formulas for counting runs of length i in binary strings

Established connections between runs in binary strings and compositions of n

Provided multiple computational methods for the problem

Abstract

We give three different computations of the total number of runs of length $i$ in binary $n$-strings, and we discuss the connection of this problem with the compositions of $n$.