# Optimal Regular Expressions for Permutations

## TL;DR

This paper constructs an optimal regular expression for permutation languages, providing explicit bounds on its size and proving its minimality among all such expressions.

## Contribution

It introduces a divide-and-conquer method to explicitly construct and prove the minimal size of regular expressions for permutation languages.

## Key findings

- Constructed a regular expression $R_n$ for permutation language $P_n$
- Derived explicit bounds on the size of $R_n$
- Proved $R_n$ has minimal size among all regular expressions for $P_n$

## Abstract

The permutation language $P_n$ consists of all words that are permutations of a fixed alphabet of size $n$. Using divide-and-conquer, we construct a regular expression $R_n$ that specifies $P_n$. We then give explicit bounds for the length of $R_n$, which we find to be $4^n n^{-(\lg n)/4+\Theta(1)}$, and use these bounds to show that $R_n$ has minimum size over all regular expressions specifying $P_n$.