bibkey: archer2024pattern authors: Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, Millie Jeske year: 2024 title: “Pattern-restricted cyclic permutations with a pattern-restricted cycle form” doi: 10.48550/arXiv.2408.15000 url: https://arxiv.org/abs/2408.15000v1 claim: “we conjecture that |A°_n(4123;1324)| is the (n−2)nd Tetranacci number, that |A°_n(2431;1324)| is the (n−1)st Pell number, and |A°_n(4132;1324)| is the (3n)th Padovan number.” strata_touched:

• D5/S3/Combinatorics/ArcherCyclicTetranacci
• D5/S3/Combinatorics/ArcherCyclicPadovan license: citation-only triage: anchor

Archer, Borsh, Bridges, Graves and Jeske, cyclic permutations with a restricted cycle form

The paper studies cyclic permutations that avoid a pattern in one-line notation while every one of their cycle forms avoids a second pattern, enumerates several such classes, and closes with conjectured enumerations for three pattern pairs.

Verified locator

DOI: 10.48550/arXiv.2408.15000

URL: https://arxiv.org/abs/2408.15000v1

• Locator: Section 1, A°_n(σ; τ) is the set of cyclic permutations of [n] whose one-line notation avoids σ and all of whose cycle forms avoid τ; a cyclic permutation has n cycle forms, the rotations of its cycle word.
• Locator: Section 4, Open Questions: “we conjecture that |A°_n(4123;1324)| is the (n−2)nd Tetranacci number, that |A°_n(2431;1324)| is the (n−1)st Pell number, and |A°_n(4132;1324)| is the (3n)th Padovan number.”

Reading of the statement

For n = 1, 2, … the three counts are 1, 1, 2, 4, 8, 15, 29, 56, 108, … for (4123; 1324), 1, 1, 2, 5, 12, 29, 70, 169, 408, … for (2431; 1324) and 1, 1, 2, 5, 12, 28, 65, 151, 351, … for (4132; 1324): the Tetranacci numbers of OEIS A000078 at index n + 2, the Pell numbers, and the Padovan numbers of OEIS A000931 at index 3n.

Bounded prior-resolution evidence

Read on 2026-09-29: the Semantic Scholar citation list of arXiv:2408.15000 has three entries. Pan, arXiv:2409.17482, proves the Pell case (2431; 1324). Pan, arXiv:2505.02045, settles two conjectures of the earlier paper arXiv:2312.05145 about the standard cycle form. An arXiv search for cyclic permutations with pattern-restricted cycle forms up to March 2026 found no treatment of (4123; 1324) or (4132; 1324). This is a bounded negative finding.