On the density of certain languages with $p^2$ letters

TL;DR

This paper explores the mathematical structure of a sequence related to language density and group actions, revealing connections between language combinatorics and algebraic group quotients for prime numbers.

Contribution

It establishes a general correspondence between language densities with $p^2$ letters and quotients of algebraic group actions for prime $p$, unifying different mathematical interpretations.

Findings

The sequence $x_n$ equals the density of certain languages with four letters.

A connection is shown between language density and quotients of $( ext{Z}_p imes ext{Z}_p)^n$ under group actions.

The paper generalizes the relationship for prime numbers $p$.

Abstract

The sequence $(x_n)_{n\in\mathbb N} = (2,5,15,51,187,\dots)$ given by the rule $x_n=(2^n+1)(2^{n-1}+1)/3$ appears in several seemingly unrelated areas of mathematics. For example, $x_n$ is the density of a language of words of length $n$ with four different letters. It is also the cardinality of the quotient of $(\mathbb Z_2\times \mathbb Z_2)^n$ under the left action of the special linear group $\mathrm{SL}(2,\mathbb Z)$. In this paper we show how these two interpretations of $x_n$ are related to each other. More generally, for prime numbers $p$ we show a correspondence between a quotient of $(\mathbb Z_p\times\mathbb Z_p)^n$ and a language with $p^2$ letters and words of length $n$.