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The Penultimate Value of the Original Word

Abstract

The return equations determine the penultimate entry.

Theorem 1.1 (A forced penultimate value).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaCube231SecondReturn.penultimate_value

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaCube231SecondReturn.penultimate_value (✓ 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 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 and its third inverse image equals itself, then its penultimate value is n minus three.

References