# The asymptotic number of prefix normal words

## TL;DR

This paper establishes the asymptotic count of prefix normal binary words of length n, revealing their exponential growth with a subexponential correction, and analyzes the maximum number of words sharing a fixed prefix normal form.

## Contribution

It provides the first precise asymptotic enumeration of prefix normal words and bounds the number of words with a given prefix normal form.

## Key findings

- Number of prefix normal words of length n is 2^{n - Θ((log n)^2)}.
- Maximum number of words with a fixed prefix normal form is 2^{n - O(√(n log n))}.
- Shows the growth rate and structural properties of prefix normal words.

## Abstract

We show that the number of prefix normal binary words of length $n$ is $2^{n-\Theta((\log n)^2)}$. We also show that the maximum number of binary words of length $n$ with a given fixed prefix normal form is $2^{n-O(\sqrt{n\log n})}$.