Stable rank-one permutations of [n]² of cycle type (2,…,2): the count table, a degree-eight polynomial, and the thirty-seven patterns of two transpositions
Abstract
A permutation u of the discrete square [n]² is stable of rank one when u ⊗ 1 and 1 ⊗ u commute in S([n]³); by a theorem of Conti and Szymański the stable permutations are exactly those whose induced endomorphism of the Cuntz algebra Oₙ is an automorphism. Brenti, Conti and Nenashev computed that [4]² carries 60 stable rank-one permutations of cycle type (2,2) and 144 of cycle type (2,2,2) and proposed the classification of the stable rank-one permutations of cycle type (2,…,2) as a direction for further research. Let N(n,k) be the number of stable rank-one permutations of [n]² of cycle type 2ᵏ. We compute N(n,k) for twenty new pairs (n,k) — among them the whole n=4 column up to the fixed-point-free case (294, 180, 324, 72, 249 for k=4,…,8) and the values N(5,5)=8500, N(6,3)=48420, N(7,3)=938910 — and prove that the two-transposition column is the degree-eight polynomial N(n,2)=(n⁸-16n⁷+114n⁶-468n⁵+1179n⁴-1780n³+1466n²-496n)/8. The mechanism is that stability depends only on the restriction of u to the indices it moves and is invariant under the diagonal action of Sₙ, so every stable permutation of cycle type 2ᵏ is the inflation of a full-support pattern on [m]² with m ≤ 4k; for k=2 there are exactly 37 patterns up to that action, and we list them. Every count asserted by a theorem here is the value of a decision procedure checked in the Lean 4 proof assistant, running as compiled code, and the structural statements are proved in Lean on top of Mathlib without any appeal to evaluation; five further values, asserted by no theorem, are marked as such.
Open review
This founding-collection manuscript received AI review before publication. Independent human review is open. Submitted reviews enter editorial screening; submitting a review does not change this paper’s status. Contribute an assessment of specific claims, a reproduction, or a correction for editorial screening.
Archived files
- Version 1 · current (opens in a new tab)
Source snapshot 2026-09-07 03:53 UTC
File fingerprint
5422a88e419504eb8e233178d694b9250a791560efdd0445cb141feaed7ce85f
Claim ledger
Stated results
CS1known data2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS2known data2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS3known data2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS4candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS5candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS6candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS7candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS8candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS9candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS10candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS11candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS12candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS13candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS14candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS15candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS16candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS17known2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS18candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS19routine2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS20routine2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS21routine2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS22routine2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS23routine2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
CS24candidate2026-09-07
Statement description is not included in this imported ledger snapshot. See the PDF for the full theorem wording.
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Brenti, Conti and Nenashev, *Hypercube subgroups of (outer) reduced Weyl groups of the Cuntz algebras* (arXiv:2601.13952v1, math.OA; v1 is the only version, arXiv API checked 2026-09-07), end with Section 7, "Outlook". Verbatim, from the arXiv PDF (page 48):
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7