Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings

TL;DR

This paper develops exact formulas, recurrences, and generating functions for counting runs in binary strings, considering various constraints and connections to compositions, with applications to probabilistic models.

Contribution

It provides a comprehensive enumeration framework for runs in binary strings, including new explicit formulas and connections to known sequences and probabilistic scenarios.

Findings

Derived recurrences and generating functions for run counts

Connected run enumeration to compositions and OEIS sequences

Extended results to probabilistic models with Bernoulli variables

Abstract

We give recurrences, generating functions and explicit exact expressions for the enumeration of fundamental quantities involving runs in binary strings. We first focus on enumerations concerning runs of ones, and we then analyse the same enumerations when runs of ones and runs of zeros are jointly considered. We give the connections between these two types of run enumeration, and with the problem of compositions. We also analyse the same enumerations with a Hamming weight constraint. We discuss which of the many number sequences that emerge from these problems are already known and listed in the OEIS. Additionally, we extend our main enumerative results to the probabilistic scenario in which binary strings are outcomes of independent and identically distributed Bernoulli variables.