Counting Finite Languages by Total Word Length

TL;DR

This paper explores the enumeration of finite languages based on total word length, providing explicit formulas, asymptotic analysis, and a Gaussian limit law for the number of words.

Contribution

It introduces an explicit counting formula, asymptotic results, and a probabilistic limit law for finite languages by total word length.

Findings

Derived explicit expression for the counting sequence.

Established asymptotic behavior for large alphabet and total length.

Proved a Gaussian limit law for the number of words in a random finite language.

Abstract

We investigate the number of sets of words that can be formed from a finite alphabet, counted by the total length of the words in the set. An explicit expression for the counting sequence is derived from the generating function, and asymptotics for large alphabet respectively large total word length are discussed. Moreover, we derive a Gaussian limit law for the number of words in a random finite language.