# A Self-Referential Property of Zimin Words

• Published in 2016
In the collections
This paper gives a short overview of Zimin words, and proves an interesting property of their distribution. Let $L_q^m$ to be the lexically ordered sequence of $q$-ary words of length $m$, and let $T_n(L_q^m)$ to be the binary sequence where the $i$-th term is $1$ if and only if the $i$-th word of $L_q^m$ encounters the $n$-th Zimin word, $Z_n$. We show that the sequence $T_n(L_q^m)$ is an instance of $Z_{n+1}$ when $1 < n$ and $m=2^n-1$.

### BibTeX entry

@article{ASelfReferentialPropertyofZiminWords,
title = {A Self-Referential Property of Zimin Words},
abstract = {This paper gives a short overview of Zimin words, and proves an interesting
property of their distribution. Let {\$}L{\_}q^m{\$} to be the lexically ordered
sequence of {\$}q{\$}-ary words of length {\$}m{\$}, and let {\$}T{\_}n(L{\_}q^m){\$} to be the binary
sequence where the {\$}i{\$}-th term is {\$}1{\$} if and only if the {\$}i{\$}-th word of {\$}L{\_}q^m{\$}
encounters the {\$}n{\$}-th Zimin word, {\$}Z{\_}n{\$}. We show that the sequence {\$}T{\_}n(L{\_}q^m){\$}
is an instance of {\$}Z{\_}{\{}n+1{\}}{\$} when {\$}1 < n{\$} and {\$}m=2^n-1{\$}.},
url = {http://arxiv.org/abs/1611.01061v1 http://arxiv.org/pdf/1611.01061v1},
year = 2016,
author = {John Connor},
comment = {},
urldate = {2021-03-31},
archivePrefix = {arXiv},
eprint = {1611.01061},
primaryClass = {math.CO},
collections = {combinatorics,fun-maths-facts}
}