LX Cayley-graph diameter

Determine the exact diameter of the directed Cayley graph of S_n generated by the left cyclic shift L and the transposition X=(1,2), proving that it equals (3n^2−8n+9)/4 for odd n and (3n^2−8n+12)/4 for even n.

Background

The LX problem originates in Glushkov’s 1968 work and has been studied extensively, but the paper states that the precise diameter has not yet been determined. The authors propose a parity-dependent quadratic formula.

References

The "LX" diameter is $(3n2-8n+9)/4$ for $n$ odd, and $(3n2-8n+12)/4$ for $n$ even (\href{https://oeis.org/A039745}{OEIS-A039745}).

— CayleyPy Growth: Efficient growth computations and hundreds of new conjectures on Cayley graphs (Brief version)  (2509.19162 - Chervov et al., 23 Sep 2025) in Section 12, subsection “Definition, diameter conjecture and related works”