{
  "schema_version": 1,
  "problem_number": "OWR-16164-019",
  "title": "A Counterexample to a Conjecture of Sportiello on Coloured Permutations in Convex Shapes",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "At the 2018 Oberwolfach workshop on enumerative combinatorics, A. Sportiello compared two families of objects attached to a digitally convex shape λ (a convex polyomino): the permutations whose graph lies in λ and that avoid a 123-pattern whose two corner cells also lie in λ, counted by A_λ, and the red–blue coloured permutations in λ that avoid two coloured versions of the 12-pattern, counted by B_λ. For the full square these numbers are the Catalan number and the central binomial coefficient, and he conjectured that B_λ ≥ A_λ for every digitally convex shape. We show that the conjecture is false. For the band λ = {(x,y) ∈ [7]² : −3 ≤ y−x ≤ 4} one has A_λ = 536 and B_λ = 515. An exhaustive computation shows that every digitally convex shape of side at most 6 satisfies the inequality, and that among the 5,693,968 shapes of side 7 exactly this band and its transpose violate it. For bands of side 10 the ratio B_λ/A_λ drops to about 0.18. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "permutation patterns",
    "coloured permutations",
    "convex polyominoes",
    "pattern avoidance",
    "counterexamples",
    "Oberwolfach Reports",
    "OWR-16164-019",
    "open mathematics",
    "mathematical proof",
    "math.CO"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-16164-019/",
  "pdf_url": "https://eulersolve.org/papers/owr-16164-019/paper.pdf?v=2492010d5c78",
  "doi": "10.5281/zenodo.23006715",
  "zenodo_record_url": "https://zenodo.org/records/23006715",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
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    "source.zip": {
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}