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Explicit Exceptional Orbits

Abstract

Three explicit orbit words leave the avoidance layers at successive depths.

Theorem 1.1 (The exceptional orbit chain).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateExceptional.cyclic_last_endpoint

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateExceptional.cyclic_last_endpoint (✓ 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 three, let C be two through n followed by one and D be n followed by two through n minus one and then one. The fundamental images of C and D are respectively n followed by one through n minus one and C; D avoids 132 through depth two, and P of C followed by n plus one does so as well. For n at least four, let E be P of the increasing word of size n minus two. Its image is D; E lies at depth five but not six, D at depth four but not five, and C at depth three but not four.

References