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Necessary Gaps at Ascents

Abstract

Avoidance of 1132 and 2213 restricts the gaps between consecutive downsteps at ascents.

Theorem 1.1 (Necessity of the gap condition).

Lean statement: D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLowGap.ascent_forces_good_gap

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLowGap.ascent_forces_good_gap (✓ 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 avoid 1132 and 2213, have Dyck shape d, and have the same distinct-letter order p at its upsteps and downsteps. If consecutive downsteps correspond to an ascent of p, then the intervening sequence is empty or is a single upstep with the preceding prefix having one more upstep than downstep.

References