Enumeration for 1233 and 1322

Abstract

The generating function for nonnesting permutations avoiding 1233 and 1322 satisfies the stated quadratic equation.

Theorem 1.1 (The 1233 and 1322 generating function).

$$claim1233$$

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

Resolves. Problems/elizalde-luo-nonnesting-1233-1322 (proved) by D5/S3/Combinatorics/Nonnesting/NonnestingRoyalHigh.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 1233 and 1322 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/NonnestingRoyalHigh.result
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalBijection
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalHookBlocks
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalIncConverse
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalReverse
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalShape
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingRoyalAvoidanceTransfer
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingRoyalLow