Systematic Enumeration of Fundamental Quantities Involving Runs in Binary Strings

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.

💡 Research Summary

The paper presents a unified combinatorial framework for counting a wide variety of run‑related statistics in binary strings. A “run” is defined according to Mo​od’s criterion – a maximal block of identical bits bounded by the opposite bit or by the string ends – and the authors explicitly allow zero‑length runs (null runs) to obtain a complete theory. With this definition they introduce families of runs such as (k ≤ ℓ)‑runs (runs whose length lies between k and ℓ), (≥ k)‑runs, and (≤ k)‑runs, and they treat runs of ones and runs of zeros both separately and jointly.

The core of the work is the enumeration of n‑length binary strings that contain exactly m runs of a given type. For the basic case of (k ≤ ℓ)‑runs the authors first derive a necessary existence condition
0 ≤ m ≤ ⌊(n+1)/(k+1)⌋,
and introduce the slack variable e = n − (mk + m − 1) ≥ 0, which measures the unused bits after placing the mandatory runs and separating zeros. A detailed recurrence (Equation 7) is obtained by conditioning on the length i of the first run: if i∈