Research
Word Structures is a research programme in combinatorics on words: the study of repetition, order and structure in sequences of symbols. Its central question concerns abelian squares — two adjacent blocks of equal length that hold the same letters the same number of times, in any order — and how far a word can go while avoiding them.
The open question
Over three letters, a verified finite example reaches 25,379 letters while avoiding every abelian square whose two halves each have length at least two.
The verified words came from V. Keränen and A. Gavrilenko; this project checked them rather than discovered them.
A finite example, however long, is still finite. It does not show that an infinite word exists.
If the rule is weakened so that only abelian squares whose halves have length six or more must be avoided, an infinite ternary word is known.
What remains open is whether an infinite ternary word can avoid them already from half-length two onward — equivalently, whether its only abelian squares can be the single-letter repetitions aa, bb and cc. This question is attributed to Mäkelä in the literature.
What is known
- Verified, finite
- A supplied ternary word of 25,379 letters checked exhaustively and found free of abelian squares at every half-length K ≥ 2. A finite witness, not an existence proof.
- Known, infinite
- An infinite ternary word avoiding abelian squares of half-length K ≥ 6 is established in the published literature.
- Open
- Whether an infinite ternary word can avoid them from half-length 2 onward — the full Mäkelä question — is unresolved.
Recorded in the claim ledger as rows 40 (the finite example), 7 (the K ≥ 6 result) and 4 (the open problem).
How we investigate it
The open range is approached from several directions at once, rather than through a single method:
- Morphisms. Substitution rules that build long or infinite words from a small seed, and whether their images stay free of abelian squares as they grow.
- Unfavourable factors. Finite blocks that cannot themselves be extended into any longer abelian-square-free word, found and proved by exhausting their possible continuations.
- Structural inspection. Close, factor-by-factor reading of specific words and morphism images to see exactly where and why a pattern appears or fails to.
- Computational extension. Searching for longer verified examples and testing where known constructions reach their limit.
- Verification. Independently checking every computed result before it can enter the claim ledger.
Current research tools
Interactive modules in the Explorer, for readers who want to work with the material directly.
Primary
Secondary
Project roles
Word Structures was founded by Joonas Huhta and Veikko Keränen. Joonas Huhta leads the project and its technical development. Veikko Keränen contributes mathematical research, develops computational methods, and provides mathematical review.
Open collaboration
Word Structures is an open research project. Contributions are welcome from researchers, developers, educators, students, and curious independent contributors.
Ways to contribute include mathematics, computational experiments, verification, software, documentation, explanations, testing, and research tools. You do not need to belong to a university or research institution to contribute.
All research claims and public conclusions remain subject to the project's evidence, verification, and review process.