The Second Semi-Meander Diagonal

Abstract

Connected semi-meanders with n crossings and winding n minus four have the second-diagonal count for every n at least four.

Definition 1.1 (Noncrossing upper arches).

$$U_n=\{M\in \mathrm{Fin}\left(2\cdot n\right)\to \mathrm{Fin}\left(2\cdot n\right)\mid \left(\forall x\in \mathrm{Fin}\left(2\cdot n\right),M\left(M\left(x\right)\right)=x\wedge M\left(x\right)\ne x\right)\wedge \left(\forall a\in \mathrm{Fin}\left(2\cdot n\right),\forall b\in \mathrm{Fin}\left(2\cdot n\right),\neg \left(a<b\wedge \left(b<M\left(a\right)\wedge M\left(a\right)<M\left(b\right)\right)\right)\right)\}$$

Formalization. D5/S3/Combinatorics/SemiMeanderSecondDiagonal/Model.UpperMatching (✓ std3).

Source. Repository-derived.

Acknowledgement. Hunter Hogan (2026). OEIS A400429, semi-meanders by crossings and winding number. URL: https://oeis.org/A400429.

Commentary.

Endpoints are numbered from zero to 2n minus one. An upper matching is a fixed-point-free involution M: each endpoint has one distinct partner, and no two upper arches have alternating endpoints a < b < M(a) < M(b). The fixed lower matching is the rainbow r(x) = 2n - 1 - x.

Remark 1.2 (Midpoint winding).

$$L_n:\mathrm{Fin}\left(n\right)\to \mathrm{Fin}\left(2\cdot n\right),\forall x\in \mathrm{Fin}\left(n\right),\mathrm{val}\left(\left(L_n\right)\left(x\right)\right)=\mathrm{val}\left(x\right),\mathrm{winding}\left(M\right)=\left|\{x\in \mathrm{Fin}\left(n\right)\mid n\le \mathrm{val}\left(M\left(\left(L_n\right)\left(x\right)\right)\right)\}\right|$$

Source. Repository-derived.

Acknowledgement. Hunter Hogan (2026). OEIS A400429, semi-meanders by crossings and winding number. URL: https://oeis.org/A400429.

Commentary.

L_n embeds Fin n into Fin(2n) without changing the endpoint value. The winding counts left-half endpoints whose upper partner lies in the right half. Each upper arch crossing the midpoint contributes exactly once.

Remark 1.3 (One loop with the lower rainbow).

$$\forall u\in \mathrm{Fin}\left(2\cdot n\right),\forall v\in \mathrm{Fin}\left(2\cdot n\right),\left(R_M\right)\left(u,v\right)\iff \left(M\left(u\right)=v\vee 2\cdot n-1-\mathrm{val}\left(u\right)=\mathrm{val}\left(v\right)\right),\mathrm{oneLoop}\left(M\right)\iff \left(\forall x\in \mathrm{Fin}\left(2\cdot n\right),\forall y\in \mathrm{Fin}\left(2\cdot n\right),\mathrm{RTC}\left(R_M,x,y\right)\right)$$

Source. Repository-derived.

Acknowledgement. Hunter Hogan (2026). OEIS A400429, semi-meanders by crossings and winding number. URL: https://oeis.org/A400429.

Commentary.

The step relation R_M joins x to y when either M(x) = y or r(x) = y, with r(x) = 2n - 1 - x. Its reflexive transitive closure connects every ordered pair of endpoints exactly when the union of upper arches and the fixed lower rainbow is one loop.

Theorem 1.4 (Exact second-diagonal count).

$$\forall n\in \mathrm{Nat},4\le n\Rightarrow \left|\{M\in U_n\mid \mathrm{winding}\left(M\right)=n-4\wedge \mathrm{oneLoop}\left(M\right)\}\right|=\frac{n^2+2\cdot n+n\mathrm{mod}2-20}{2}$$

Proof. Machine-checked in Lean as D5/S3/Combinatorics/SemiMeanderSecondDiagonal.result (✓ std3). ∎

Resolves. Problems/oeis-a400429-semi-meander-second-diagonal (proved) by D5/S3/Combinatorics/SemiMeanderSecondDiagonal.result.

Source. Repository-derived.

Acknowledgement. Hunter Hogan (2026). OEIS A400429, semi-meanders by crossings and winding number. URL: https://oeis.org/A400429.

Commentary.

For every n at least four, count exactly the noncrossing upper matchings whose midpoint winding is n minus four and whose union with the lower rainbow is one loop. The second diagonal of OEIS A400429 is this set: its source index k = floor(n/2) - 1 gives winding n - 4. At n = 4 the count is two. Di Francesco, Golinelli and Guitter predicted this same polynomial from the resummation in Appendix D (1996); the count here is an independent exact proof for the stated model and full range.

References

• Truth anchor: D5/S3/Combinatorics/SemiMeanderSecondDiagonal.result
• Truth anchor: D5/S3/Combinatorics/SemiMeanderSecondDiagonal/Model.UpperMatching
• Dependency: D5/S3/Combinatorics/SemiMeanderSecondDiagonal/Stage5