Quasi-polynomial diameters of symmetric and alternating groups

Establish that, for any algorithmically constructible generating sets of the symmetric group S_n or alternating group A_n, the Cayley-graph diameter is eventually given by a quadratic or linear quasi-polynomial in n, potentially with a common leading term among all constituents.

Background

The paper reports extensive computational evidence that diameters of many Cayley graphs of S_n and A_n follow quadratic or linear quasi-polynomial formulas, with the constituent polynomial depending on n modulo a fixed integer. The authors formulate an explicitly labeled “Extremely optimistic” conjecture for generating sets that can be produced in polynomial time.

References

For any generators of $S_n$ (or $A_n$) which can be constructed by an algorithm with say polynomial complexity in $n$ (e.g. a Python function which takes as input $n$ and outputs generators in time polynomial in $n$) the diameter of the Cayley graph will be given by some quadratic or linear quasi-polynomial in $n$ (at least for $n$ large enough). Even more optimistically, the leading terms of all constituents coincide.

— 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 3, subsection “Diameter quasi-polynomiality conjecture”