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Counting Parameters with a Final Maximum

Abstract

Parameters ending at their maximum are counted by a smaller family beginning at its maximum.

Theorem 1.1 (The final-maximum parameter count).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateEndpointCount.last_endpoint_count

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateEndpointCount.last_endpoint_count (✓ 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 n at least two, the number of parameter permutations of size n plus one ending with n plus one whose P images avoid 132 through depth two equals one plus the number of depth-two avoiders of size n beginning with n.

References