Enumeration of simple permutations sortable by two parallel pop stacks with bypass

Determine the number of simple permutations of each size that can be sorted by two pop stacks in parallel when entries may bypass both pop stacks, and establish whether the sequence satisfies the stated Fibonacci formulas: a₀ = a₁ = 1, a₂ = 2, aₙ = F₂ₙ₋₅ − 1 for odd n ≥ 3, and aₙ = F₂ₙ₋₅ for even n > 3.

Background

The paper studies permutation-sorting devices built from pop stacks augmented with a bypass operation. Earlier, it establishes the sortable-permutation class for two pop stacks in parallel with bypass and proves that this class has a rational generating function. The authors then focus on the subclass of simple permutations within that sortable class.

The proposed enumeration concerns the number aₙ of simple permutations of size n sortable by two parallel pop stacks with bypass. The conjectured values are expressed in terms of Fibonacci numbers and are presented as a possible closed form for the corresponding sequence.

References

However, there is also a potential nice closed form for the simple sortable permutations that we boldly put here as a conjecture: 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 $n\geq 3$ is odd, and $a_n =F_{2n-5}$ if $n > 3$ is even (where $F_n$ is the $n$-th Fibonacci number).

— Sorting permutations using a pop stack with a bypass  (2503.08285 - Cioni et al., 11 Mar 2025) in Section 7, “Further work,” subsection “Conjecture on the enumeration of simple permutations sortable by two pop stacks in series”