Shape of Parameters with a Final Maximum
Abstract
The final-maximum family consists of two explicit words and a cyclic family.
Theorem 1.1 (The three possible shapes).
Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateEndpointShape.last_endpoint_shape
Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateEndpointShape.last_endpoint_shape (✓ 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 a stem permutation r of size h at least three, P of r followed by h plus one avoids 132 through depth two exactly when r is increasing, r is two through h followed by one, or r begins with h and ends with one and its inverse fundamental image avoids 132 through depth two.
References
- Truth anchor:
D5/S3/Combinatorics/FundamentalBijection/ThetaIterateEndpointShape.last_endpoint_shape - Dependency: D5/S3/Combinatorics/FundamentalBijection/ThetaBasicInverseBlocks
- Dependency: D5/S3/Combinatorics/FundamentalBijection/ThetaBasicInverseGeneral
- Dependency: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateCycleReduction
- Dependency: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateExceptional
- Dependency: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateLastEndpoint
- Dependency: D5/S3/Combinatorics/FundamentalBijection/ThetaIteratePositionBlocks
- Dependency: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateTail