Shuffle Squares in Differentiable Words

Abstract

Experiments on binary run-length differentiability lead to sharp computer-assisted criteria for shuffle squares. Every C³-word of length greater than $16$ is a shuffle square exactly when both letter multiplicities are even. All $34$ nonempty even-Parikh exceptions are smooth and persist in every higher differentiability class. For C² the sharp threshold is $48$, with $212$ nonempty exceptions. At level C¹ no global parity threshold exists, but every even-Parikh non-shuffle-square of length at least $36$ has proper nonempty shuffle-square prefixes and suffixes. Exactly $230$ nonempty even-Parikh C¹-words have no nonempty shuffle-square prefix. The full tree avoiding such prefixes eventually consists of $422$ periodic rays. Consequently, a nonempty Kolakoski prefix is a shuffle square exactly when both multiplicities are even, apart from lengths $4$ and $8$. We also characterize classes of morphisms reflecting shuffle squares. For doubly binary words, we determine the exact deletion distance and largest twins, and prove sharp bounds for single local repairs. Exact recurrences, residual-state checks, and separate Python programs make the finite computations reproducible.

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Cite

BibTeX RIS

BibTeX

@misc{zwierzynski2026shuffle,
  title = {{Shuffle Squares in Differentiable Words}},
  author = {Michał Zwierzyński},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07828},
  eprint = {2610.07828},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07828},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Shuffle Squares in Differentiable Words
AU  - Michał Zwierzyński
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2610.07828
UR  - https://arxiv.org/abs/2610.07828
AB  - Experiments on binary run-length differentiability lead to sharp computer-assisted criteria for shuffle squares. Every C^3-word of length greater than $16$ is a shuffle square exactly when both letter multiplicities are even. All $34$ nonempty even-Parikh exceptions are smooth and persist in every higher differentiability class. For C^2 the sharp threshold is $48$, with $212$ nonempty exceptions. At level C^1 no global parity threshold exists, but every even-Parikh non-shuffle-square of length at least $36$ has proper nonempty shuffle-square prefixes and suffixes. Exactly $230$ nonempty even-Parikh C^1-words have no nonempty shuffle-square prefix. The full tree avoiding such prefixes eventually consists of $422$ periodic rays. Consequently, a nonempty Kolakoski prefix is a shuffle square exactly when both multiplicities are even, apart from lengths $4$ and $8$. We also characterize classes of morphisms reflecting shuffle squares. For doubly binary words, we determine the exact deletion distance and largest twins, and prove sharp bounds for single local repairs. Exact recurrences, residual-state checks, and separate Python programs make the finite computations reproducible.
ER  -

Abstract

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