Reflection of Dyck Paths

Abstract

Reversal with exchanged step directions is an involution on Dyck paths.

Definition 1.1 (Reflected path equivalence).

Lean statement: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalReverse.reflection

Formalization. D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalReverse.reflection (✓ 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.

Reverse a Dyck path and exchange upsteps with downsteps. This operation gives a bijection of Dyck paths with itself, with the same operation as inverse.

References

• Truth anchor: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalReverse.reflection
• Dependency: D5/S3/Combinatorics/Nonnesting/NonnestingBasicRoyalEncoding