Diameter of the LRX Cayley graph

Prove that the diameter of the Cayley graph of the symmetric group S_n generated by the left cyclic shift L, right cyclic shift R, and transposition X=(0,1) is n(n-1)/2 for every n.

Background

The paper studies the LRX Cayley graph of S_n, whose generators are the two cyclic shifts and the transposition of the first two positions. Computations for small n and machine-learning experiments support the conjectured diameter n(n-1)/2, but the authors prove only lower and upper bounds around this value. Consequently, establishing the exact diameter remains unresolved.

References

Surprisingly, according to \href{https://oeis.org/A186783}{OEIS-A186783}, it is an open conjecture that the diameter of the Cayley graph is $n(n-1)/2$ for $S_n$.

— CayleyPy RL: Pathfinding and Reinforcement Learning on Cayley Graphs  (2502.18663 - Chervov et al., 25 Feb 2025) in Section 1, subsection “Present paper: approach, benchmarks, mathematical theorems and conjectures”; Section 5.1, “LRX Cayley graph”