Cyclic Permutations Avoiding 4132 with 1324-Avoiding Cycle Forms

Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves and Millie Jeske, Pattern-restricted cyclic permutations with a pattern-restricted cycle form, arXiv:2408.15000v1, Section 4:

we conjecture that … |A°_n(4132;1324)| is the (3n)th Padovan number

Here A°_n(4132; 1324) is the set of cyclic permutations of 1, …, n whose one-line notation avoids 4132 and all of whose cycle forms, the rotations of the cycle word, avoid 1324.

Let P_0 = 1, P_1 = P_2 = 0 and P_{k+3} = P_{k+1} + P_k (OEIS A000931). The theorem D5/S3/Combinatorics/ArcherCyclicPadovan.result establishes |A°_n(4132; 1324)| = P_{3n} for every n ≥ 1; the counts for n = 1, …, 9 are 1, 1, 2, 5, 12, 28, 65, 151, 351. Equivalently, their generating function is x(1 − x)² / (1 − 3x + 2x² − x³).

Pre-registration issue 11215 records the literature screen. Of the three papers citing arXiv:2408.15000, Pan's arXiv:2409.17482 proves only the Pell case (2431; 1324) and Pan's arXiv:2505.02045 settles conjectures of the earlier paper arXiv:2312.05145 on the standard cycle form. The repository had no result for this class. These are bounded negative findings.

1. Encode a cyclic permutation by its cycle word rooted at 1. A selected quadruple is a 1324 in some rotation exactly when the three entries read cyclically after its minimum form a 213.
2. Write a valid cycle word as 1, L, 2, R. If L is empty, deleting 2 gives a valid word of length n − 1. Otherwise every entry of L exceeds every entry of R, R is increasing, and 1, L standardizes to a valid word.
3. In that block form the one-line word avoids 4132 exactly when its tail avoids 4132 and the tail entries below the first entry of L increase. This defines an auxiliary class D_k with d_k = b_k + d_{k−1}, and b_n = b_{n−1} + Σ_{k=1}^{n−2} d_k.
4. Eliminating d gives b_n = 3b_{n−1} − 2b_{n−2} + b_{n−3} for n ≥ 4, the recurrence satisfied by P_{3n}, with matching initial values 1, 1, 2.

The statement would fail if some n had a count different from P_{3n}. It depends on reading "all cycle forms" as every rotation of the cycle word and on the Padovan indexing of OEIS A000931.

Exhaustive enumeration of all cycle words through n = 10 agrees with the decomposition, both one-line criteria, the auxiliary split and the counts 1, 1, 2, 5, 12, 28, 65, 151, 351, 816.

theorem; the conjecture is stated in the Open Questions of arXiv:2408.15000v1 and is quantified over every n.

The literature screen is limited to the citation list, arXiv searches and repository checks recorded above.