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bibkey: difrancesco1996meanders authors: P. Di Francesco, O. Golinelli, E. Guitter year: 1996 title: “Meanders: A Direct Enumeration Approach” doi: 10.1016/S0550-3213(96)00505-6 url: https://arxiv.org/abs/hep-th/9607039 claim: “Appendix D (D.14) gives a resummed small-t prediction whose second diagonal yields the A400429 polynomial; the text does not establish its all-n coefficient validity.” strata_touched:

  • D5/S3/Combinatorics/SemiMeanderSecondDiagonal license: citation-only triage: anchor

Di Francesco, Golinelli and Guitter, meanders

P. Di Francesco, O. Golinelli and E. Guitter, Meanders: A Direct Enumeration Approach, Nuclear Physics B 482 (1996), 497-535, DOI 10.1016/S0550-3213(96)00505-6, arXiv:hep-th/9607039. The 46-page arXiv PDF at https://arxiv.org/pdf/hep-th/9607039 was inspected on 29 September 2026, especially Appendix D, PDF pages 41-44 (printed pages 40-43), and the conclusion, PDF page 32 (printed page 31).

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  • DOI: 10.1016/S0550-3213(96)00505-6 (Nuclear Physics B 482, 497-535).
  • URL: https://arxiv.org/abs/hep-th/9607039 (46-page arXiv PDF, Appendix D and conclusion inspected 2026-09-29).

Appendix D defines a generating function in (D.1) whose q exponent counts connected components and whose t exponent indexes the winding deficit: t^j corresponds to winding n-2j. For fixed k, (D.9) expresses the count with n-k components and winding n-2j as a polynomial in n, with small-n corrections. The authors say they computed these polynomials for 0<=j<=k<=14 from enumeration through n<=24. The text then says, “we expect the numbers” to be polynomials “for large enough n”. Equations (D.11)-(D.13) concern large-n and large-q asymptotics.

Before (D.14), the authors write that they “have been able to re-sum the large q series coefficients of this expansion up to order 3 in t”. The displayed t^2 rational terms of the two functions in (D.14), inserted into the asymptotic factorization (D.11), predict the same second diagonal polynomial (n^2+2*n+(n mod 2)-20)/2. The conclusion calls (D.14) a “purported re-summation”. The inspected text does not prove that extracting the q^1 t^2 coefficient of this asymptotic resummation is valid for every finite n: that coefficient has k=n-1, outside the fixed-k scope of (D.9). Thus the polynomial was predicted in 1996; it is not a new formula, and the present theorem is described as an independent all-n proof rather than a certified first proof.