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

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 $n$.

arXiv comment10 pages, 2 figures. The section on cobordism categories was shortened. http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i3p16