LX Glushkov diameter conjecture

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

Background

This is the paper’s proposed resolution of a diameter problem originating with V. M. Glushkov in 1968. The precise diameter had not been determined despite substantial prior study, and the conjectured formula is fitted to computed growth data.

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”