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The Shortest Interesting Binary Words

Gabriele Fici

TL;DR

This work argues that the binary words $v=0011$ and $w=001011$ are the shortest binary words exhibiting a breadth of nontrivial combinatorial properties, acting as central test cases across palindromicity, repetitions, Lyndon/de Bruijn theory, and factor complexity. By examining anti-palindromicity, palindromic length, richness, square/overlap behavior, morphic images (e.g., Thue–Morse), Pansiot coding, de Bruijn generalizations, and scattered-subword structure, the paper situates these two words as foundational building blocks in combinatorics on words. The results illuminate why these words appear in many papers and how their extremal properties provide insight into classical concepts such as Lyndon factorizations, de Bruijn sequences, and minimal forbidden factors. Overall, the study establishes a cohesive narrative linking these short words to a wide spectrum of combinatorial phenomena, underscoring their significance as the shortest interesting binary words.

Abstract

I will show that there exist two binary words (one of length 4 and one of length 6) that play a special role in many different problems in combinatorics on words. They can therefore be considered \textit{the shortest interesting binary words}. My claim is supported by the fact that these two words appear in dozens of papers in combinatorics on words.

The Shortest Interesting Binary Words

TL;DR

This work argues that the binary words and are the shortest binary words exhibiting a breadth of nontrivial combinatorial properties, acting as central test cases across palindromicity, repetitions, Lyndon/de Bruijn theory, and factor complexity. By examining anti-palindromicity, palindromic length, richness, square/overlap behavior, morphic images (e.g., Thue–Morse), Pansiot coding, de Bruijn generalizations, and scattered-subword structure, the paper situates these two words as foundational building blocks in combinatorics on words. The results illuminate why these words appear in many papers and how their extremal properties provide insight into classical concepts such as Lyndon factorizations, de Bruijn sequences, and minimal forbidden factors. Overall, the study establishes a cohesive narrative linking these short words to a wide spectrum of combinatorial phenomena, underscoring their significance as the shortest interesting binary words.

Abstract

I will show that there exist two binary words (one of length 4 and one of length 6) that play a special role in many different problems in combinatorics on words. They can therefore be considered \textit{the shortest interesting binary words}. My claim is supported by the fact that these two words appear in dozens of papers in combinatorics on words.
Paper Structure (5 sections, 10 theorems, 9 equations, 1 table)

This paper contains 5 sections, 10 theorems, 9 equations, 1 table.

Table of Contents

  1. Introduction
  2. Palindromes and Anti-palindromes
  3. Squares and Other Repetitions
  4. Lyndon and de Bruijn Words
  5. Factors and Scattered Subwords

Key Result

Proposition 1

The derivative of an anti-palindrome is always a palindrome. On the opposite, the derivative of a palindrome is never an anti-palindrome.

Theorems & Definitions (16)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • Proposition 6
  • ...and 6 more