LRX diameter conjecture

Prove that the diameter of the Cayley graph of S_n generated by the left cyclic shift, right cyclic shift, and transposition (1,2) equals n(n−1)/2.

Background

The LRX generating set is obtained by adjoining both cyclic shifts to the transposition (1,2), making the generating set inverse-closed. The paper identifies the exact quadratic diameter formula as a known but unresolved conjecture.

References

This conjecture is still open, currently best upper/lower bounds and further results were proposed in .

— 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”

OEIS-A186783 which presents a conjecture that the diameter for LRX generators is $n(n-1)/2$. This conjecture is still open, currently best upper/lower bounds and further results were proposed in .

— 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”