A Descending Parameter at the First Endpoint

Abstract

A maximal first letter and a terminal one force a descending parameter when its successor word avoids 132.

Theorem 1.1 (The descending permutation).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateFirstEndpoint.descending_of_last_one_and_b_avoids

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateFirstEndpoint.descending_of_last_one_and_b_avoids (✓ 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 permutation of size at least two beginning with its maximum and ending with one is decreasing if its successor word b avoids 132.

References

• Truth anchor: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateFirstEndpoint.descending_of_last_one_and_b_avoids
• Dependency: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateCycle