# A Self-Referential Property of Zimin Words

- Published in 2016
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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 $m=2^n-1$.

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- key
- ASelfReferentialPropertyofZiminWords
- type
- article
- date_added
- 2021-03-31
- date_published
- 2016-04-10

### BibTeX entry

@article{ASelfReferentialPropertyofZiminWords, key = {ASelfReferentialPropertyofZiminWords}, type = {article}, title = {A Self-Referential Property of Zimin Words}, author = {John Connor}, 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{\$}.}, comment = {}, date_added = {2021-03-31}, date_published = {2016-04-10}, urls = {http://arxiv.org/abs/1611.01061v1,http://arxiv.org/pdf/1611.01061v1}, collections = {combinatorics,fun-maths-facts}, url = {http://arxiv.org/abs/1611.01061v1 http://arxiv.org/pdf/1611.01061v1}, year = 2016, urldate = {2021-03-31}, archivePrefix = {arXiv}, eprint = {1611.01061}, primaryClass = {math.CO} }