bibkey: archer2024fundamental authors: Kassie Archer, Robert P. Laudone year: 2024 title: “Pattern avoidance and the fundamental bijection” doi: 10.48550/arXiv.2407.06338 url: https://arxiv.org/abs/2407.06338v1 claim: “For σ ∈ {231, 312}, the generating functions F_σ^k(x) are rational.” strata_touched:
- D5/S3/Combinatorics/FundamentalBijection/ThetaCube
- D5/S3/Combinatorics/FundamentalBijection/ThetaIterate license: citation-only triage: anchor
Archer and Laudone, pattern avoidance and the fundamental bijection
The paper studies permutations avoiding a pattern of length three whose images under the fundamental bijection, and under its iterates, also avoid that pattern, and permutations fixed by an iterate of the fundamental bijection; it closes with several conjectured enumerations.
Verified locator
DOI: 10.48550/arXiv.2407.06338
URL: https://arxiv.org/abs/2407.06338v1
- Locator: Section 2, the fundamental bijection θ writes a permutation in standard cycle form, each cycle beginning with its largest element and the cycles ordered by increasing largest element, and erases the parentheses.
- Locator: Section 4, Conjecture 4.5: for n ≥ 2, t_n^2(132) = k^3 + 3k^2 + 2k − 1 (n = 3k), k^3 + 4k^2 + 4k (n = 3k + 1), k^3 + 5k^2 + 7k + 2 (n = 3k + 2), with the values 3n − 4, 2n − 1, n + 2 and 5 for k = 3, 4, 5 and k ≥ 6, where t_n^k(σ) counts the π with π, θ(π), …, θ^k(π) all avoiding σ.
- Locator: Section 5, Conjecture 5.6: for σ ∈ {231, 312}, F_σ^3(x) = 1/(1 − x − x^2 − 2x^3), F_σ^4(x) = 1/(1 − x − x^2 − 2x^4 − x^5 − x^6) and F_σ^5(x) = 1/(1 − x − x^2), where F_σ^k counts the σ-avoiders fixed by θ^k.
Reading of the statement
For σ ∈ {231, 312} the numbers of σ-avoiders fixed by θ^3 are 1, 1, 2, 5, 9, 18, 37, 73, 146, 293, 585 for n = 0, …, 10. The numbers t_n^2(132) for n = 1, …, 10 are 1, 2, 5, 9, 15, 23, 32, 44, 59, 75.
Bounded prior-resolution evidence
Read on 2026-09-30: the Semantic Scholar citation list of arXiv:2407.06338 contains arXiv:2312.05145, arXiv:2408.15000, arXiv:2505.05651 and arXiv:2607.21466; none treats Conjectures 4.5 or 5.6. The journal version, Electron. J. Combin. 32(4) (2025) P4.19, still states both as conjectures. This is a bounded negative finding.