A Self-Referential Property of Zimin Words

TL;DR

This paper explores a unique property of Zimin words, demonstrating a specific self-referential pattern in their distribution within lexicographically ordered q-ary words, revealing a recursive structure related to Zimin words.

Contribution

It proves a novel property linking the distribution of Zimin words in lexicographic sequences to a recursive pattern, expanding understanding of their combinatorial structure.

Findings

Sequence T_n(L_q^m) is an instance of Z_{n+1} when 1 < n and m=2^n-1

Establishes a self-referential property of Zimin words in lexicographic order

Provides insight into the recursive structure 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 $m=2^n-1$.