Padovan Enumeration of Cyclic Avoiders
Abstract
The cyclic avoidance numbers for 4132 and 1324 follow a subsequence of the Padovan numbers.
Theorem 1.1 (The Padovan enumeration).
Proof. Machine-checked in Lean as D5/S3/Combinatorics/ArcherCyclicPadovan.result (✓ std3). ∎
Resolves. Problems/archer-cyclic-4132-1324-padovan (proved) by D5/S3/Combinatorics/ArcherCyclicPadovan.result.
Source. Repository-derived.
Acknowledgement. Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, Millie Jeske (2024). Pattern-restricted cyclic permutations with a pattern-restricted cycle form. DOI: 10.48550/arXiv.2408.15000. URL: https://arxiv.org/abs/2408.15000v1.
Commentary.
For every positive n, the number of cyclic permutations avoiding 4132 in one-line form and 1324 in every cycle form equals the Padovan number at index three times n.
References
- Truth anchor:
D5/S3/Combinatorics/ArcherCyclicPadovan.result - Dependency: D5/S3/Combinatorics/ArcherCyclicPadovanArithmetic
- Dependency: D5/S3/Combinatorics/ArcherCyclicPadovanCount
- Dependency: D5/S3/Combinatorics/ArcherCyclicPadovanCycleWords