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Cycle Compatibility of Insertion

Abstract

Insertion preserves the distinguished-cycle avoidance condition for inverse parameters.

Theorem 1.1 (The inserted cycle and its image).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateInsertionCycle.insertion_cycle

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateInsertionCycle.insertion_cycle (✓ 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.

If q is a permutation of size h at least two beginning with h and ending with one, let r be its inverse fundamental image. The image of I(r) is h plus three, followed by q increased by one, followed by h plus two and one, and the inverse image of this word is I(r). Moreover, P(I(r)) avoids 132 through depth two exactly when P(r) does.

References