slug: archer-cyclic-4123-1324-tetranacci bibkey: archer2024pattern doi: 10.48550/arXiv.2408.15000 url: https://arxiv.org/abs/2408.15000v1 triage: theorem motivation_gids:

• D5/S3/Combinatorics/ArcherCyclicTetranacci.result

Cyclic Permutations Avoiding 4123 with 1324-Avoiding Cycle Forms

Problem

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(4123;1324)| is the (n−2)nd Tetranacci number

Here A°_n(4123; 1324) is the set of cyclic permutations of 1, …, n whose one-line notation avoids 4123 and all of whose cycle forms, the rotations of the cycle word, avoid 1324.

Motivation

Let T_0 = T_1 = T_2 = 0, T_3 = 1 and T_{k+4} = T_{k+3} + T_{k+2} + T_{k+1} + T_k (OEIS A000078). The theorem D5/S3/Combinatorics/ArcherCyclicTetranacci.result establishes |A°_n(4123; 1324)| = T_{n+2} for every n ≥ 1; the counts for n = 1, …, 9 are 1, 1, 2, 4, 8, 15, 29, 56, 108.

Gap

Pre-registration issue 11214 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.

Route

1. Encode a cyclic permutation by its cycle word rooted at 1. Four entries give a 1324 in some rotation exactly when their circular order can be read as x, y, z, t with x < z < y < t; hence the part of a valid cycle word after 1 avoids 213.
2. If 2 does not follow 1 directly, write the word as 1, P, 2, S. Every entry of P exceeds every entry of S, S is increasing, P begins with the smallest entry above S, and one-line 4123-avoidance forces |S| ≤ 2.
3. The maps that insert a new entry after 1, or a block 2, 3, …, k with k = 2, 3, 4, preserve both conditions in both directions, and every valid word arises from exactly one of them.
4. The resulting recurrence a_n = a_{n−1} + a_{n−2} + a_{n−3} + a_{n−4} with a_1 = a_2 = 1, a_3 = 2, a_4 = 4 matches T_{n+2}.

Falsifier

The statement would fail if some n had a count different from T_{n+2}. It depends on reading “all cycle forms” as every rotation of the cycle word; requiring only the standard cycle form to avoid 1324 gives a different class.

Evidence

Exhaustive enumeration of all cycle words through n = 10 agrees with every structural lemma and with the counts 1, 1, 2, 4, 8, 15, 29, 56, 108, 208.

Triage

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

ASSUMED-UNVERIFIED

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