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The Initial Pair after the Maximum and One

Abstract

A prescribed return entry forces the next two values of a 231-avoider.

Theorem 1.1 (The next two initial values).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaCube231Shape.return_edge_forces_initial_pair

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaCube231Shape.return_edge_forces_initial_pair (✓ 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 231-avoiding permutation of size n at least five starting with n, one and ending with n minus one, if its inverse image has two at position n minus three, then its next two entries are n minus two and two.

References