Enumeration for 1132 and 2213

Abstract

The generating function for nonnesting permutations avoiding 1132 and 2213 satisfies the stated quadratic equation.

Theorem 1.1 (The 1132 and 2213 generating function).

$$claim1132$$

Proof. Machine-checked in Lean as D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLow.result (✓ std3). ∎

Resolves. Problems/elizalde-luo-nonnesting-1132-2213 (proved) by D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLow.result.

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 R be the ordinary generating function counting doubled nonnesting permutations avoiding 1132 and 2213 by the number of distinct letters. Then xR squared - (1 - x) squared times R + (1 - x) squared equals zero.

References

• Truth anchor: D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLow.result
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalDownRuns
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalGround
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalGroundSeries
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingDefs
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLowFiber