Avoidance from Permitted Gaps
Abstract
The increasing-block order and permitted Dyck gaps imply avoidance of 1132 and 2213.
Theorem 1.1 (Sufficiency of the gap condition).
Lean statement: D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLowConverse.good_gaps_avoid_low
Proof. Machine-checked in Lean as D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLowConverse.good_gaps_avoid_low (✓ 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 a doubled nonnesting word have Dyck shape d and the same distinct-letter order p at its upsteps and downsteps. Suppose p avoids 132 and 213. Whenever consecutive downsteps in d correspond to an ascent of p, require either that they are adjacent or that they enclose a single upstep with the preceding prefix having one more upstep than downstep. Then the word avoids 1132 and 2213.
References
- Truth anchor:
D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLowConverse.good_gaps_avoid_low - Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalBijection
- Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalBlocks
- Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalShape
- Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLowGap