Deletion of the Least Value

Abstract

Deleting the smallest doubled letter preserves avoidance and first order.

Theorem 1.1 (Deleting both ones preserves avoidance).

Lean statement: D5/S3/Combinatorics/Nonnesting/NonnestingBasicDeletion.delete_lowest_avoider

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Nonnesting/NonnestingBasicDeletion.delete_lowest_avoider (✓ 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 both copies of one and decrementing the remaining letters sends an avoider of size n plus one to an avoider of size n.

Theorem 1.2 (First order survives filtering).

Lean statement: D5/S3/Combinatorics/Nonnesting/NonnestingBasicDeletion.first_order_after_filter

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Nonnesting/NonnestingBasicDeletion.first_order_after_filter (✓ 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.

Filtering a word above k preserves the order of first occurrences of any two retained letters.

Truth anchor: D5/S3/Combinatorics/Nonnesting/NonnestingBasicDeletion.delete_lowest_avoider
Truth anchor: D5/S3/Combinatorics/Nonnesting/NonnestingBasicDeletion.first_order_after_filter
Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicSum