{
  "schema_version": 1,
  "problem_number": "OWR-14298158-012",
  "title": "A Proper Subclass of the Separable Permutations Whose Generating Function Lies Outside Q(z, √(1−4z), √(1−6z+5z²))",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Question 5 of R. Brignall's abstract (joint work with M. Bóna, C. Defant, M. Opler and V. Vatter) in Oberwolfach Report 6/2024, p. 292, asks whether every proper subclass of the separable permutations Av(2413, 3142) has a generating function in K = Q(z, √(1−4z), √(1−6z+5z²)). The answer is no. The class C = Av(2341, 2413, 3142) is a proper subclass of the separable permutations which contains Av(231). Its generating function f = Σ_{n≥0} |C_n| z^n satisfies z(1−z+z²)f² − (1−2z+2z²)f + (1−z)² = 0, so that f = (1−2z+2z² − √D)/(2z(1−z+z²)) with D = 1−8z+20z²−24z³+16z⁴−4z⁵. This enumeration is due to Callan, Mansour and Shattuck (Discrete Math. Theor. Comput. Sci. 19:1 (2017), #6, Theorem 11), and the same generating function is in the PermPAL database. The polynomial D is irreducible over Q and coprime to (1−4z)(1−z)(1−5z), whereas a nonzero element of Q(z) is a square in K only if it lies in the square class of 1, 1−4z, (1−z)(1−5z) or (1−4z)(1−z)(1−5z); hence √D ∉ K and f ∉ K. The contribution of this note is the observation that this class answers Question 5 negatively, a self-contained proof of its enumeration by the substitution decomposition, and an exact certificate that its generating function is not in K. This is an unrefereed note.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "separable permutations",
    "permutation classes",
    "pattern avoidance",
    "algebraic generating functions",
    "Av(2341, 2413, 3142)",
    "substitution decomposition",
    "square classes",
    "Kummer theory",
    "counterexample",
    "Oberwolfach Report 6/2024",
    "UnsolvedMath",
    "OWR-14298158-012",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-16",
  "publication_date": "2026-10-16",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-16",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-14298158-012/",
  "pdf_url": "https://eulersolve.org/papers/owr-14298158-012/paper.pdf?v=a4eeea5e16e4",
  "doi": "10.5281/zenodo.23421224",
  "zenodo_record_url": "https://zenodo.org/records/23421224",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The enumeration of Av(2341, 2413, 3142) is due to Callan, Mansour and Shattuck (2017) and is in PermPAL; the note contributes the observation that this class answers Question 5, a self-contained proof of the enumeration and an exact certificate that its generating function is not in the field.",
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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."
}