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Decomposition of the Second Avoidance Layer

Abstract

The second avoidance layer splits into a cycle family and two exceptional permutations.

Theorem 1.1 (The families beginning or not ending at the maximum).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateDecomposition.second_layer_decomposition

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateDecomposition.second_layer_decomposition (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Kassie Archer, Robert P. Laudone (2024). Pattern avoidance and the fundamental bijection. DOI: 10.48550/arXiv.2407.06338. URL: https://arxiv.org/abs/2407.06338v1.

Commentary.

For size n at least four, the second-layer avoiders beginning with n are the image under P of eligible parameter permutations of size n minus two together with n, two, three, up to n minus one, one; their number is the parameter count plus one. The second-layer avoiders not ending with n consist of this family together with two, three, up to n, one; their number is the parameter count plus two.

References