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Recursive Form of Increasing-Order Words

Abstract

Removing both ones leaves one of two insertion forms.

Theorem 1.1 (Two reconstruction cases).

Lean statement: D5/S3/Combinatorics/Nonnesting/NonnestingFourRecursive.increasing_reconstruct

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Nonnesting/NonnestingFourRecursive.increasing_reconstruct (✓ 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.

An increasing-order avoider of size n plus one is either the direct sum of 11 and its shifted smaller word or arises by inserting 121 before a shifted tail.

References