bibkey: blanco2025prefixreversals authors: Saúl A. Blanco, Mikhail P. Golubyatnikov, Elena V. Konstantinova, Natalia V. Maslova, Luka A. Nikiforov year: 2025 title: ‘Generating the symmetric group by three prefix reversals’ doi: 10.48550/arXiv.2511.16959 claim: Conjecture 4 describes a triangular region of index triples whose three prefix reversals do not generate the symmetric group. strata_touched:

• D5/S0/CayleyGrowth/PrefixReversalTripleOddNonGeneration
• D5/S0/CayleyGrowth/PrefixReversalTripleEvenNonGeneration license: citation-only triage: anchor

Generating the symmetric group by three prefix reversals

The cubic pancake graphs are Cayley graphs of the symmetric group generated by three prefix reversals, where the prefix reversal of length L sends x to L + 1 - x for x at most L and fixes every larger point. Characterising the triples that generate the whole group is open; the paper settles the four families in which one of the two smallest or two largest lengths appears, and records five conjectures.

Conjecture 4 is the triangular non-generating region, and reads: if m + k < n - 1 then the three reversals of lengths n, m, k generate a proper subgroup; and if n is even and m + k < n, the same conclusion holds. The standing domain is 2 <= k < m < n. The condition m + k appears in no theorem of the paper, only in this conjecture; the version of 1 September 2026 still states it as open.

The repository declaration proves the first clause for odd n. Collect the pairs actually exchanged by the three reversals into a graph on the positions. A pair coming from length L has endpoints summing to L + 1, so the three families are disjoint and the edge count is the sum of the three halved lengths rounded down. Each generator moves every position either to itself or along a single edge, so every element of the generated subgroup maps each connected component onto itself; a transposition across two components then lies outside the subgroup. For odd n the hypothesis forces at most n - 2 edges, while a connected graph on n vertices needs at least n - 1, so the graph is disconnected and the subgroup is proper.

The even clause is proved in the companion module D5/S0/CayleyGrowth/PrefixReversalTripleEvenNonGeneration. At that boundary the graph can be connected, and the connected branch is real rather than hypothetical: among even triples with n < 200 satisfying the hypothesis, 7105 have exactly n - 1 edges and 5301 of those are connected, the smallest being n = 6, m = 3, k = 2. So counting alone does not settle the even case. When the graph is connected it is a tree, and giving the edges of the longest reversal weight one and the others weight zero, then labelling each position by the weight parity of its unique path from a fixed root, splits the positions into two halves that the two shorter reversals preserve and the longest exchanges. A transposition across the halves then lies outside the subgroup.

With both modules the conjecture is settled in full: the second clause covers even n, and the first clause at even n follows from it because m + k < n - 1 implies m + k < n.

The two strategies used here are the ones the paper names for its own non-generation results: an invariant set of positions, and a nontrivial block system. The contribution is that they settle the whole triangular region for odd n, which the paper leaves open, not that either strategy is new.

Search log

• 2026-09-13: Read arXiv:2511.16959v2 in full. Conjecture 4 appears in Section 4.1 with the statement quoted above, and m + k occurs nowhere else in the paper.
• 2026-09-13: Searched the pinned Mathlib checkout and the repository for prefix reversals and pancake graphs. No match in either, so the permutation and the auxiliary graph are defined here. Mathlib/Combinatorics/SimpleGraph/Acyclic.lean supplies Connected.card_vert_le_card_edgeSet_add_one, which carries the counting step and is used directly.
• 2026-09-13: Checked the conjecture numerically from the defining permutations. Every triple with n at most nine satisfying either clause generates a proper subgroup, ten in all; outside the region fourteen of seventeen triples generate the whole group, so the check is not vacuous.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2511.16959