Square-with-whiskers pattern for extremal generators

Prove that generators producing maximal or nearly maximal diameters for S_n and A_n follow the “square with whiskers” pattern for infinitely many n, for both directed and undirected Cayley graphs.

Background

The paper introduces a graphical representation of permutation generators and identifies a recurring pattern consisting of one 4-cycle and two branches. Computational results through n≤15 support the claim that this pattern governs generators with unusually large diameters.

References

The generators with maximal (or nearly) diameter for $S_n$ and $A_n$ follow the "square with whiskers" pattern (at least for infinite number of $n$). We expect that to be true for both undirected and directed Cayley graphs.

— 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 6, subsection “Graphical representation for any generators -- pattern ‘square with whiskers’”