# The number of languages with maximum state complexity

## TL;DR

This paper provides a formula for counting the number of finite languages with maximum state complexity and generalizes the concept from languages to functions on finite sets.

## Contribution

It introduces a formula for the number of maximum-complexity languages and extends the analysis from languages to functions on finite sets.

## Key findings

- Derived a formula for counting maximum-complexity languages.
- Generalized the maximum complexity analysis from languages to functions.
- Enhanced understanding of the distribution of maximum-complexity languages.

## Abstract

C\^{a}mpeanu and Ho (2004) determined the maximum finite state complexity of finite languages, building on work of Champarnaud and Pin (1989). They stated that it is very difficult to determine the number of maximum-complexity languages. Here we give a formula for this number. We also generalize their work from languages to functions on finite sets.