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The Penultimate Record Block

Abstract

A third-iterate return fixes an entry near the end of the inverse image.

Theorem 1.1 (Two near the end of the inverse image).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaCube231Return.penultimate_block_ends_two

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaCube231Return.penultimate_block_ends_two (✓ 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 greater than three starting with n, one, ending with n minus one, and fixed by the third inverse iterate, the entry of its inverse image at position n minus three is two.

References