Downstep-Weighted Dyck Enumeration

Abstract

Adjacent downsteps determine a weight on Dyck paths.

Definition 1.1 (Downstep weight sum).

Lean statement: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalDownCount.downWeight

Formalization. D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalDownCount.downWeight (✓ 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.

For semilength n, the downstep weight sum is the sum over all Dyck paths of two to the power of their number of adjacent downstep pairs.

References

• Truth anchor: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalDownCount.downWeight
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalDownRuns