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Insertion of Three Letters

Abstract

A three-letter insertion preserves two avoidance conditions and has a unique inverse on its specified boundary shape.

Definition 1.1 (The three-letter insertion).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateInsertion.I

Formalization. D5/S3/Combinatorics/FundamentalBijection/ThetaIterateInsertion.I (✓ 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 parameter of length h, I places h plus three and h plus two first, increases each parameter letter after the first by one, and finishes with one and h plus one.

Theorem 1.2 (The successor word and avoidance after insertion).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateInsertion.insertion_scan

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateInsertion.insertion_scan (✓ 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 first-maximum parameter r of length h at least two, I(r) is a permutation of size h plus three. Its successor word is h plus two, the first h minus one entries of b(r) increased by one, one, the last entry of b(r) increased by one, and h plus three. Avoidance of 132 by r and by b(r) is respectively equivalent to avoidance by I(r) and by b(I(r)). Every permutation of size h plus three beginning with h plus three, h plus two and ending with one, h plus one has a unique first-maximum parameter preimage under I.

References