RotationAvoidanceSymmetry

Abstract

Complement and reverse preserve the cardinalities of permutation classes defined by avoidance in the first k rotations.

Theorem 1.1 (Wilf equivalence within an orbit).

Lean statement: D5/S3/Combinatorics/RotationAvoidance/RotationAvoidanceSymmetry.orbit_wilfEquivalent

Proof. Machine-checked in Lean as D5/S3/Combinatorics/RotationAvoidance/RotationAvoidanceSymmetry.orbit_wilfEquivalent (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Ömer Eğecioğlu, Collier Gaiser, Mei Yin (2026). Pattern avoidance in permutations and their rotations. DOI: 10.48550/arXiv.2607.20750. URL: https://arxiv.org/abs/2607.20750v1.

Commentary.

For every positive integer k, every permutation q of one through four and every pattern s in the complement and reverse orbit of q, the patterns q and s are Wilf-equivalent for k rotations. Thus S_n^(k)(q) and S_n^(k)(s) have equal cardinalities for every n at least k.

References

• Truth anchor: D5/S3/Combinatorics/RotationAvoidance/RotationAvoidanceSymmetry.orbit_wilfEquivalent
• Dependency: D5/S3/Combinatorics/RotationAvoidance/RotationAvoidanceDefs