**The question**

What is the order of magnitude for the function $n\mapsto |{\rm cd}(S_n)|$?

**The motivation**

In my research on character degrees of finite groups, I have in recent years been focusing on symmetric groups as a test bed for general conjectures. Among my more recent investigations, I have come up with (what is to me, at least) an interesting reduction: I can prove a statement true regarding the full set ${\rm cd}(S_n)$ of degrees of irreducible, complex-valued characters on the symmetric group of degree $n$ (for a fixed $n$) by showing a statement is true regarding the set of primes $\{p\leq n\mid p~{\rm is~prime}\}$. What I would like is to develop a feeling for just how much of a reduction this is. Obviously, the prime counting function is approximated by $n/\ln n$, but I don't have a feel for, as a function of $n$, how quickly the function $|{\rm cd}(S_n)|$ grows. I have a reduction in effort *to* a sublinear function, and I would like to know what that reduction is *from*. Is it quadratic, is it exponential?

**The (failed) searches**

As an aside, I admit that part of my problem is knowing how to conduct the research. For example, if I do a Google search for "how quickly does |cd(Sn)| grow", I get results pertaining to cancer. If I add "math:" at the beginning, the returns are from freshmen calculus involving convergence of sequences. When I write things out in words using no symbols, other issues become evident. Such searches, when correctly limited to math, are further complicated by the dual use of the word "degree" in this context, the degree of the permutation group versus the degrees of the representations. Furthermore, results abound on the maximal norm of character values and asymptotics related to those values. But, results on the order of magnitute for the function $n\mapsto |{\rm cd}(S_n)|$ are at the very least ~~very well hidden~~. (OK, not so much; a comment below points out there is an OEIS entry).

1more comment