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The Balanced Two-Block Family

Abstract

The balanced decreasing-block word satisfies the required avoidance conditions.

Theorem 1.1 (Avoidance by the balanced word and its images).

Lean statement: D5/S3/Combinatorics/FundamentalBijection/ThetaIterateWTests.W_tests

Proof. Machine-checked in Lean as D5/S3/Combinatorics/FundamentalBijection/ThetaIterateWTests.W_tests (✓ 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 h at least four, W(h) permutes one through h, and W(h), b(W(h)), and the fundamental image of W(h) all avoid 132; consequently P(W(h)) avoids 132 through depth two.

References