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Cycle Reduction at a Final Maximum

Abstract

A final-maximum parameter reduces to the inverse image of its stem.

Theorem 1.1 (The stem and its inverse image).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateLastEndpoint.last_endpoint_cycle

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateLastEndpoint.last_endpoint_cycle (✓ 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 r of size h at least two beginning with h, depth-two avoidance of P of r followed by h plus one forces r to end with one. If r ends with one, its inverse fundamental image begins with h and has fundamental image r, and the P construction avoids through depth two exactly when that inverse image does.

References