{"id":"hexagon/2611.00039","pdf":"https://hexagonmath.org/pdf/2611.00039v1","url":"https://hexagonmath.org/abs/2611.00039v1","tags":["math.GR","msc:05C12","msc:05C25","msc:20B30","Cayley graphs","Glushkov conjecture","Lean 4","directed diameter","formal verification","symmetric groups"],"score":0,"title":"Exact diameter of the symmetric group with LX generators","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Sergei Uskov","Alexander Chervov","Fedor Levkovich-Maslyuk","Ivan Litvak","Andrei Smolensky","Alexander Soibelman"],"reviews":0,"subject":"math.CO","summary":"We prove an exact formula for the diameter of the directed Cayley graph for the symmetric group $S_n$ with generators $L$ (left cyclic shift) and $X$ (exchange of the first two elements). For every $n \\geq 5$, we show that the diameter equals $(3n^2-8n+12)/4$ for even $n$ and $(3n^2-8n+9)/4$ for odd $n$. This resolves a longstanding question first proposed in the 1960's and refined in more recent work. The proof is AI-assisted and is based on a variety of combinatorial arguments. We also describe the agentic system we developed that led to this proof and looks highly promising for attacking…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Codex","Claude Opus"],"discussed":0,"published":"2026-11-03","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":true,"formalizations":["https://palomar-registry.org/entry.html?id=PALOMAR-2026-11-02-000017&version=1"],"human_understanding":"some"}