Thanks a lot for your comment, I think it can be an answer. Sorry if this is not a research-level question.
I have this practical algorithm and I'm trying to express its running time as a function of some parameters and not only the size of the input. If parametrized-complexity is not the right keyword, could you help me to understand what is parametrized-complexity and to find the right keyword?
Following what you said, I have: O(ppp!(np)k)=O(ep√p⋅pknk)=O(1√k(e⋅nk)k). If now I replace p by k in the original formula, I have O(nkk!).
So I can claim (i) the complexity of the algorithm is in O((n4)k), (ii) the complexity of the algorithm is in O(1√k(e⋅nk)k) and (iii) the complexity of the algorithm is in O(nkk!). Is that correct?