Character-degree invariants for symmetric groups

Determine whether the system of invariants consisting of the order n(G) and the indicator tuple δ^{cd}(G) for the set of complex irreducible character degrees is full for the class of symmetric groups.

Background

The paper proves that the order together with the set of element orders or conjugacy class sizes distinguishes symmetric groups among finite groups. It states that the corresponding recognition problem based on the set of irreducible character degrees remains unresolved, while the multiplicity-sensitive character-degree invariant is known to suffice.