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Transfer Between Two Avoidance Classes

Abstract

Reversed occurrence positions transfer avoidance between the two pairs of patterns.

Theorem 1.1 (Avoidance under occurrence reversal).

Lean statement: D5/S3/Combinatorics/Nonnesting/NonnestingRoyalAvoidanceTransfer.avoidance_transfer

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Nonnesting/NonnestingRoyalAvoidanceTransfer.avoidance_transfer (✓ 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.

Let positive block sizes sum to n. Let w and v be doubled nonnesting words whose first-occurrence orders are respectively the increasing-block permutation and the hook-block permutation with reversed block sizes. Suppose, for each index i less than n, the first position of the reversed-index hook letter in v plus the second position of the corresponding increasing-block letter in w plus one equals the length of w, and the same equality holds with first and second positions exchanged. Then w avoids 1132 and 2213 if and only if v avoids 1233 and 1322.

References