bibkey: cioni2025sorting authors: Lapo Cioni, Luca Ferrari, Rebecca Smith year: 2025 title: “Sorting permutations using a pop stack with a bypass” doi: 10.1016/j.disc.2025.114964 url: https://arxiv.org/abs/2503.08285v1 claim: “Simple permutations sortable by two pop stacks in parallel with bypass are counted by F_{2n-5} - 1 (n odd) and F_{2n-5} (n even).” strata_touched:

• D5/S3/Combinatorics/PopStack/PopStackSimple license: citation-only triage: anchor

Cioni, Ferrari and Smith, sorting permutations using a pop stack with a bypass

The paper studies permutations sortable by a pop stack whose entries may bypass the stack, and by several such machines in series and in parallel, determines bases of the sortable classes, and closes with open problems on their enumeration.

Verified locator

DOI: 10.1016/j.disc.2025.114964

URL: https://arxiv.org/abs/2503.08285v1

• Locator: Section 9, the permutations sortable by two pop stacks in parallel with bypass form the class Av(2341, 25314, 42513, 42531, 45213, 45231, 52314, 642135, 642153).
• Locator: Section 9.2, Conjecture: the number a_n of simple permutations of size n in this class satisfies a_0 = a_1 = 1, a_2 = 2, a_n = F_{2n−5} − 1 for odd n ≥ 3 and a_n = F_{2n−5} for even n > 3.

Reading of the statement

The numbers of simple permutations in the class for n = 0, …, 11 are 1, 1, 2, 0, 2, 4, 13, 33, 89, 232, 610, 1596; the class itself has 1, 1, 2, 6, 23, 97, 418, 1800, 7717, … elements (OEIS A374165).

Bounded prior-resolution evidence

Read on 2026-10-01 and 2026-10-02: arXiv lists only version 1 of the preprint. The statement appears again as Conjecture 8 in the authors’ contribution to the Permutation Patterns 2025 booklet (July 2025). The abstract of the published version (Discrete Math. 349(5), 2026) coincides with that of arXiv version 1 and does not announce an enumeration of the simple sortable permutations; the full text of the published version was not read. Searches of arXiv, the citation index and GitHub located no later treatment of the conjecture. This is a bounded negative finding.