The 12̇-Machine Sorts a Central Binomial Number of Permutations

Michael Yang, Hansen Shieh and Ashley Yu, Stack-Sorting with Dotted-Pattern-Avoiding Stacks, arXiv:2411.11914v2, Section 6, Conjecture 6.1 (printed page 10):

The machine-sortable permutations under the 12̇-machine map in Sₙ are enumerated by (2n−2 choose n−1).

The 12̇-machine is s ∘ s₁₂̇, with s West's stack-sorting map. By the source's Proposition 3.1, s₁₂̇ reverses each peak run, a peak run being a maximal block of consecutive entries that starts at a left-to-right maximum. A permutation is machine-sortable when the machine maps it to the identity.

The theorem D5/S3/Combinatorics/DottedStack/ShiehYangYuTwelveDot.result establishes the statement: for every n ≥ 1, exactly (2n − 2 choose n − 1) permutations of [n] are machine-sortable.

Pre-registration issue 12225 records the literature screen: no work citing arXiv:2411.11914 treats this conjecture, and GitHub searches for the identifier returned no hit outside this repository. This is a bounded negative finding.

1. West's criterion is proved from the stack algorithm: s(w) is increasing exactly when w avoids 231.
2. Every image of the peak-run reversal ends in n, and the preimages of σn correspond to the subsets of the record positions of σ; so the count is the sum of 2^{rec(σ)} over 231-avoiding permutations σ of [n − 1].
3. A recursive bijection with Dyck paths, through the last primitive excursion, carries records to excursions.
4. Coloring each excursion by a bit and reflecting the marked excursions gives the balanced ±1 bridges of length 2(n − 1), which are counted by (2n − 2 choose n − 1).

The statement would fail if West's criterion or the fibre description missed a sortable permutation, or if the excursion decomposition or the reflection were not invertible.

Every structural lemma, the fibre bijection and both inverse identities were checked exhaustively on all permutations of length at most 11; the counts agree with (2n − 2 choose n − 1) through n = 11.

theorem; the statement is Conjecture 6.1 of arXiv:2411.11914 and is quantified over every n ≥ 1.

• Proved (formalized): the count of machine-sortable permutations of [n] equals the sum of 2^{rec(σ)} over 231-avoiders σ of [n − 1], and equals (2n − 2 choose n − 1).
• Literature: the identity "Dyck paths of semilength m weighted by 2 to the number of returns sum to (2m choose m)" is classical; the new content is the fibre description of the 12̇-machine, which reduces the conjecture to it.
• Open: the source's companion questions for other dotted patterns beyond 12̇ and 21̇, and the distribution of the number of machine passes needed to sort.

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