# What's the relationship of $n^n$ and $n!$ complexity with respect to EXPTIME?

We know that:

• $$k^n = o(n!)$$
• $$n! = o(n^n)$$

where $$o$$ refers to "little o notation" (i.e. bounds that are asymptotically not tight).

With that in mind:

• Are there any defined or academically established classes for problems with complexity of at most $$O(n!)$$ and $$O(n^n)$$?
• What is the set relationship for these classes of problems with respect to $$\mathsf{EXPTIME}$$ or e.g. $$\mathsf{2\mbox{-}EXPTIME}$$?

$$\mathsf{EXPTIME}$$ is the class of problems solvable by algorithms whose running time is $$O(2^{n^c})$$ for some $$c>0$$. In contrast, $$\mathsf{E}$$ is the class of problems solvable by algorithms whose running time is $$O(2^{cn})$$ for some $$c>0$$.
These definitions show that problems solvable in time $$O(n!)$$ or $$O(n^n)$$ are in $$\mathsf{EXPTIME}$$, but not necessarily in $$\mathsf{E}$$.