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arxiv: 2604.23188 · v1 · submitted 2026-04-25 · 🧮 math.CO · cs.DM

Binary Words Containing Few Abelian Squares

Szilard Zsolt Fazekas , Adam Mammoliti , Robert Mercas , Jamie Simpson This is my paper

Pith reviewed 2026-05-08 07:57 UTC · model grok-4.3

classification 🧮 math.CO cs.DM
keywords abelian squaresbinary wordsParikh vectorminimal numberconjecturecombinatorics on wordssquare-free words
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The pith

For any Parikh vector over two letters, an explicit binary word is constructed that is conjectured to contain the fewest abelian squares.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Fici and Saarela conjectured that every binary word of length n contains at least floor(n/4) abelian squares. The paper extends this conjecture slightly and proves the lower bound for certain special cases. For the remaining cases it supplies an explicit construction, for each possible Parikh vector, of a binary word that is conjectured to realize the global minimum number of abelian squares. A sympathetic reader would care because these constructions would then give the exact minimal count for every binary word and would supply canonical examples with the least repetition of this kind.

Core claim

We slightly extend this conjecture and show that it holds in some special cases. In all other cases we have the following: given a Parikh vector over a two letter alphabet we produce a word with that Parikh vector which we conjecture contains the least possible number of abelian squares.

What carries the argument

The explicit construction, for each Parikh vector, of a binary word conjectured to minimize the number of abelian squares.

If this is right

  • The floor(n/4) lower bound holds for the special cases where it is proved.
  • The minimal number of abelian squares is conjectured to equal the count in the constructed word for every other Parikh vector.
  • The constructions give a concrete method to produce words that attain the conjectured minimum for any length and letter distribution.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the conjecture is true, the exact minimal number of abelian squares is now known for every binary word.
  • The constructions could be used to test related questions about the distribution or avoidance of abelian squares in binary words.
  • Small cases could be checked by exhaustive search to see whether any shorter or different word beats the construction.

Load-bearing premise

The constructed words actually achieve the smallest possible number of abelian squares among all binary words with the same Parikh vector.

What would settle it

A single binary word with a given Parikh vector that contains strictly fewer abelian squares than the word produced by the construction for that vector.

read the original abstract

Fici and Saarela ([2]) conjectured that a binary word of length n contains at least $\lfloor n/4 \rfloor$ abelian squares. We slightly extend this conjecture and show that it holds in some special cases. In all other cases we have the following: given a Parikh vector over a two letter alphabet we produce a word with that Parikh vector which we conjecture contains the least possible number of abelian squares.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends the Fici-Saarela conjecture that every binary word of length n contains at least floor(n/4) abelian squares. It proves the extended conjecture in certain special cases and, for remaining binary Parikh vectors, supplies explicit constructions of words conjectured to realize the minimal number of abelian-square factors.

Significance. The special-case proofs constitute a concrete, verifiable contribution to combinatorics on words. The explicit constructions for general Parikh vectors are a strength, as they are described in a form that permits direct checking and potential use in future work; if the minimality conjecture holds, the manuscript would supply both supporting evidence for the bound and candidate extremal examples.

major comments (2)
  1. [Special cases section] The proofs establishing the bound in special cases are internally consistent and load-bearing only for those cases; the manuscript does not claim a general proof.
  2. [General constructions section] For general Parikh vectors the constructions are presented as conjecturally minimal, yet no matching lower-bound argument or exhaustive verification ruling out a word with the same letter counts but strictly fewer abelian squares is supplied; this minimality assumption is load-bearing for any claim that the constructions support the extended conjecture beyond the special cases.
minor comments (2)
  1. The abstract and introduction could more sharply separate the proven special-case results from the conjectural general constructions to prevent misreading of the scope.
  2. Notation for Parikh vectors and abelian squares is standard but would benefit from a short reminder or forward reference in the preliminaries for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment of the special-case proofs and the explicit constructions. We respond to the major comments point by point below.

read point-by-point responses

  1. Referee: [Special cases section] The proofs establishing the bound in special cases are internally consistent and load-bearing only for those cases; the manuscript does not claim a general proof.

    Authors: We agree with this characterization. The proofs are developed only for the indicated special cases of Parikh vectors, and the manuscript explicitly refrains from claiming a general result. The comment accurately reflects the paper's scope. revision: no

  2. Referee: [General constructions section] For general Parikh vectors the constructions are presented as conjecturally minimal, yet no matching lower-bound argument or exhaustive verification ruling out a word with the same letter counts but strictly fewer abelian squares is supplied; this minimality assumption is load-bearing for any claim that the constructions support the extended conjecture beyond the special cases.

    Authors: We acknowledge that the constructions for general Parikh vectors are presented without a general lower-bound proof or exhaustive verification of minimality. This is because proving that these (or any) words achieve the minimal number of abelian squares for arbitrary Parikh vectors is the content of the extended conjecture itself, which remains open. The constructions are supplied as explicit, checkable candidates conjectured to realize the bound, thereby furnishing supporting evidence and material for future verification. The manuscript does not invoke their minimality to establish the conjecture outside the special cases already proved; the conjectural status is stated in the abstract and introduction. No revision is required. revision: no

standing simulated objections not resolved
  • A general lower-bound proof establishing that the supplied constructions (or any binary words) achieve the minimal number of abelian squares for all Parikh vectors.

Circularity Check

0 steps flagged

No circularity; explicit constructions with openly conjectural minimality.

full rationale

The paper extends the Fici–Saarela conjecture on the minimum number of abelian squares in binary words and proves the bound in special cases using direct arguments. For remaining Parikh vectors it supplies explicit word constructions and states only a conjecture that these achieve the global minimum; no derivation claims to prove minimality from the constructions, no parameters are fitted and then relabeled as predictions, and no load-bearing step reduces to a self-citation or self-defined ansatz. The logic is therefore self-contained: proven parts rest on case analysis, while the general claim is explicitly labeled conjectural rather than derived.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard definitions of abelian squares and Parikh vectors from prior literature; no free parameters, new axioms, or invented entities are introduced.

axioms (2)
  • standard math An abelian square is a factor uv where the Parikh vector of u equals the Parikh vector of v.
    Standard definition in combinatorics on words, invoked throughout the abstract.
  • domain assumption The alphabet is binary.
    The entire study is restricted to two-letter words.

pith-pipeline@v0.9.0 · 5363 in / 1265 out tokens · 43786 ms · 2026-05-08T07:57:41.795546+00:00 · methodology

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Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages

  1. [1]

    and Puzynina, S., 2023

    Fici, G. and Puzynina, S., 2023. Abelian combinatorics on words: a survey. Computer Science Review, 47, p.100532

    work page 2023

  2. [2]

    On The Minimum Number of Abelian Squares in a Word Combinatorics and Algorithmics of Strings , Dagstuhl Reports, 4(2014), Issue 3, 34-35

    Fici, Gabriel and Aleksi Saarela. On The Minimum Number of Abelian Squares in a Word Combinatorics and Algorithmics of Strings , Dagstuhl Reports, 4(2014), Issue 3, 34-35

    work page 2014

  3. [3]

    ”On weak circular squares in binary words.” In Annual Symposium on Combinatorial Pattern Matching, pp

    Fraenkel, Aviezri S., Jamie Simpson, and Mike Paterson. ”On weak circular squares in binary words.” In Annual Symposium on Combinatorial Pattern Matching, pp. 76-82. Berlin, Heidelberg: Springer Berlin Heidelberg, 1997

    work page 1997

  4. [4]

    ”Solved and unsolved problems about abelian squares.” arXiv preprint arXiv:1802.04481 (2018)

    Simpson, Jamie. ”Solved and unsolved problems about abelian squares.” arXiv preprint arXiv:1802.04481 (2018). Akita University Japan Email address:<szilard.fazekas@gmail.com> School of Mathematics and Statistics University of New South W ales NSW 2052 Australia Email address:adam.mammoliti@outlook.com.au Loughborough University UK Email address:R.G.Mercas...

    work page arXiv 2018