slug: oeis-a400429-semi-meander-second-diagonal bibkey: hogan2026a400429 doi: null url: https://oeis.org/A400429 triage: theorem motivation_gids:

• D5/S3/Combinatorics/SemiMeanderSecondDiagonal.result

OEIS A400429 second semi-meander diagonal

Problem

OEIS A400429 revision 16 states verbatim: “Conjecture: D_2(n) = (n^2 + 2*n + n mod 2 - 20)/2, for n >= 4.” Its diagonal is D_j(n)=T(n,floor(n/2)-j+1), where T(n,k) counts one-loop semi-meanders with n crossings and winding 2*(k-1)+(n mod 2). The source’s equivalent arch model fixes a lower rainbow pairing of 2*n vertices and counts noncrossing upper perfect matchings whose union with it is one loop. The second diagonal has exactly n-4 upper arches crossing the midpoint.

Motivation

The entry still labels this all-order formula a conjecture. Issue #11034 registered the external open-problem settlement before the Lean proof probe; the issue is a registration locator, not mathematical evidence.

Gap

Di Francesco, Golinelli and Guitter (1996), Appendix D (D.9)-(D.14), give a resummed small-t prediction whose second-diagonal coefficient is the same polynomial. Their conclusion calls (D.14) a “purported re-summation”. The inspected argument does not establish all finite-n coefficient validity. The formula is credited to that prediction; this result is an independent proof, with no first-proof claim.

Route

Index vertices from zero. A Lean UpperMatching n is the fixed-point-free involution on Fin (2*n) for the upper arches. Its noncrossing condition excludes a<b<M(a)<M(b). The lower rainbow is x.rev=2*n-1-x. M.winding counts left-half vertices x<n with M(x)>=n; this counts each midpoint-crossing upper arch once. M.oneLoop says every two vertices are connected by the reflexive transitive closure of upper and lower pairing steps. Therefore the theorem’s nested subtype counts precisely source objects with winding n-4 and one loop, for every natural n>=4.

The proof cuts both halves at the midpoint. Four vertices on each side belong to non-crossing local pairs; the other n-4 upper arches cross the midpoint. A connected union with the lower rainbow forces the local pairing types and their compatible cross-half connections. Their count gives (n^2+2*n+(n mod 2)-20)/2, with natural division by two. At n=4, the formula is 2, and the source table has T(4,1)=2.

Falsifier

A noncrossing upper matching with one-loop rainbow union and n-4 midpoint-crossing arches counted differently from the source’s D_2(n), or any n>=4 whose model count differs from the polynomial, would defeat the claimed source bridge or formula respectively. Bounded table agreement alone does not settle the unbounded claim.

Evidence

The Lean model is in D5/S3/Combinatorics/SemiMeanderSecondDiagonal/Model.lean; its five ordered Stage1.lean through Stage5.lean modules establish the cut classification, connectivity and count. The short D5/S3/Combinatorics/SemiMeanderSecondDiagonal.lean module states the exact all-n result. The mathematical source mapping and theorem statement were independently approved before this publication-artifact work. Source attribution, the Scribe interpretation, deposit, independent publication review and merge are separate from the kernel statement.

Triage

Tier 1 external named conjecture (OEIS A400429, revision 16), registered in issue #11034. theorem; resolution proved; admission basis open-problem-resolution; proof_shape: content. The proof classifies the local pairings and compatible cross-half connections on the live path.

ASSUMED-UNVERIFIED

The literature check covers OEIS A400429 revision 16 and the cited 1996 arXiv PDF, including Appendix D and its conclusion. It is not an exhaustive search of later literature or a priority certification. The separate declared_validated escape audit remains unfinished (issue #11180).