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Removing the Doubled Largest Prefix

Abstract

The largest-letter prefix is primitive and leaves an ordered avoider.

Theorem 1.1 (Primitivity of the largest prefix).

Lean statement: D5/S3/Combinatorics/Nonnesting/NonnestingFourPrimitiveLarge.primitive_large_prefix

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Nonnesting/NonnestingFourPrimitiveLarge.primitive_large_prefix (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Sergi Elizalde, Amya Luo (2024). Pattern avoidance in nonnesting permutations. DOI: 10.48550/arXiv.2412.00336. URL: https://arxiv.org/abs/2412.00336v6.

Commentary.

A word beginning with two copies of its largest letter has no proper value cut.

Theorem 1.2 (Tail after the largest prefix).

Lean statement: D5/S3/Combinatorics/Nonnesting/NonnestingFourPrimitiveLarge.large_first_tail

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Nonnesting/NonnestingFourPrimitiveLarge.large_first_tail (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Sergi Elizalde, Amya Luo (2024). Pattern avoidance in nonnesting permutations. DOI: 10.48550/arXiv.2412.00336. URL: https://arxiv.org/abs/2412.00336v6.

Commentary.

Removing the two initial largest letters from such an avoider leaves an avoider of size one less with increasing first-occurrence order.

References