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Incompatible Terminal Entries

Abstract

The specified terminal configuration cannot return after three inverse applications.

Theorem 1.1 (A terminal contradiction).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaCube312Final.terminal_collision

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaCube312Final.terminal_collision (✓ 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 size n at least seven, no permutation fixed by the third inverse iterate has n minus two, n minus one, two, n at positions n minus five through n minus two, while its inverse image starts with n, has n minus three at position two and two at position n minus two, and its second inverse image also starts with n.

References