A Generating Function for the Distribution of Runs in Binary Words

TL;DR

This paper derives generating functions for counting binary words with a fixed number of runs of a certain length, providing explicit formulas and extending to words with specific run counts of ones using symmetry properties.

Contribution

It introduces explicit generating functions for the distribution of runs in binary words, including extensions to counts of runs of ones, advancing combinatorial enumeration methods.

Findings

Explicit generating function for $N(n,r,0)$: $(1-x)(1-2x + x^r - x^{r+1})^{-1}

Generating function for $N(n,r,k)$: $x^{kr}$ times the $(k+1)$-th power of the base function

Extended symmetry-based methods to count words with exactly $k$ runs of ones

Abstract

Let $N(n,r,k)$ denote the number of binary words of length $n$ that begin with $0$ and contain exactly $k$ runs (i.e., maximal subwords of identical consecutive symbols) of length $r$. We show that the generating function for the sequence $N(n,r,0)$, $n=0,1,\ldots$, is $(1-x)(1-2x + x^r-x^{r+1})^{-1}$ and that the generating function for $\{N(n,r,k)\}$ is $x^{kr}$ time the $k+1$ power of this. We extend to counts of words containing exactly $k$ runs of $1$s by using symmetries on the set of binary words.