The sequence $(x_n)_{n\in\mathbb N} = (2,5,15,51,187,\ldots)$ 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 $n$.
@article{10_37236_4668,
author = {Carlos Segovia and Monika Winklmeier},
title = {On the density of certain languages with \(p^2\) letters},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {3},
doi = {10.37236/4668},
zbl = {1353.37095},
url = {http://geodesic.mathdoc.fr/articles/10.37236/4668/}
}
TY - JOUR
AU - Carlos Segovia
AU - Monika Winklmeier
TI - On the density of certain languages with \(p^2\) letters
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/4668/
DO - 10.37236/4668
ID - 10_37236_4668
ER -
%0 Journal Article
%A Carlos Segovia
%A Monika Winklmeier
%T On the density of certain languages with \(p^2\) letters
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/4668/
%R 10.37236/4668
%F 10_37236_4668
Carlos Segovia; Monika Winklmeier. On the density of certain languages with \(p^2\) letters. The electronic journal of combinatorics, Tome 22 (2015) no. 3. doi: 10.37236/4668
