Location of the Parameter Maximum
Abstract
The maximum of an eligible parameter lies at exactly one endpoint.
Theorem 1.1 (Exactly one maximal endpoint).
Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateEndpoints.parameter_endpoints
Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateEndpoints.parameter_endpoints (✓ 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.
A parameter permutation of size at least two whose P image avoids 132 through depth two has its maximum at its first or last position, but not at both.
References
- Truth anchor:
D5/S3/Combinatorics/FundamentalBijection/ThetaIterateEndpoints.parameter_endpoints - 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