Simple Permutations Sortable by Two Parallel Pop Stacks with Bypass

Lapo Cioni, Luca Ferrari and Rebecca Smith, Sorting permutations using a pop stack with a bypass, arXiv:2503.08285v1, Section 9.2:

Let a_n be the number of simple permutations of size n that can be sorted by a machine consisting of two pop stacks in parallel where entries are allowed to bypass the pop stacks. Then a_0 = a_1 = 1, a_2 = 2, a_n = F_{2n−5} − 1 if n ≥ 3 is odd, and a_n = F_{2n−5} if n > 3 is even.

The sortable permutations are the class C = Av(2341, 25314, 42513, 42531, 45213, 45231, 52314, 642135, 642153); a permutation is simple when it has no interval of length strictly between 1 and n.

The theorem D5/S3/Combinatorics/PopStack/PopStackSimple.result establishes the stated values for every n: the simple permutations of length n in C number 1, 1, 2 for n = 0, 1, 2 and F_{2n−5} − (n mod 2) for n ≥ 3.

Pre-registration issue 11689 records the literature screen: no later paper, citing record or public repository treats the conjecture. The authors restate it as Conjecture 8 in the Permutation Patterns 2025 booklet, and the abstract of the published version is unchanged from arXiv version 1. This is a bounded negative finding.

1. Intervals in C behave well under union, contraction and inflation; inserting a new minimum in one of the first three positions is characterized exactly, and a small auxiliary class of two-chain permutations with its simple and prefix-only members is described completely.
2. The simple permutations of C, together with one extra non-simple permutation for each odd length, are split by the position of their minimum: position 2, position 3, or later.
3. An explicit recursive map T with inverse and an explicit bijection Φ show that the first two parts each have b_{n−1} elements, and a continuation V with proved inverse shows that the third part has b_{n−1} − b_{n−2} elements.
4. Hence b_n = 3b_{n−1} − b_{n−2} with b_3 = 1 and b_4 = 2, so b_n = F_{2n−5}, and removing the odd extra permutation gives the stated formula.

The statement would fail if some simple permutation of C had no place in the three-part split, if T, Φ or V were not bijective, or if the odd extra permutation were simple.

Every structural lemma, the three-part split and both inverse identities of T, Φ and V were checked on all permutations of length at most 11; the counts agree through n = 11.

theorem; the statement is the conjecture of Section 9.2 of arXiv:2503.08285 and is quantified over every n.

The literature screen is limited to the arXiv, citation-index, OEIS and GitHub searches and the repository checks recorded above; the published journal version was not read in full.