Exact diameter of the symmetric group with LX generators
Source: hexagonmath.org/abs/2611.00039v1
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…
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