Largest-diameter generating sets contain involutions

Determine whether a generating set with largest diameter for S_n contains an involution for infinitely many n, in both directed and undirected Cayley-graph settings.

Background

Exhaustive searches for small n and randomized searches for somewhat larger n suggest that large diameters occur when one generator is an involution. The paper elevates this observed pattern to an explicit conjecture covering both directed and undirected cases.

References

The generating set with largest diameter for $S_n$ contains an involution (at least for infinite number of $n$), both for directed and undirected cases.

— 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 5, subsection “Distribution of diameter over conjugacy classes pairs -- involutions strikes”