One day I thought of an interesting problem :

“ Given $$$n$$$ elements that are initially $$$0$$$, operate $$$k$$$ times, each time will randomly select an element, increase it by $$$1$$$, and find the expected value of the maximum value among the $$$n$$$ elements”

I used to think it was a very simple problem (at the beginning), but actually I couldn't find any good solution (at the beginning, I only knew the an $$$O(k^2\ln k)$$$ solution)

After asking some powerful people, I learned one way to solve this problem in $$$O(\textrm{poly}~k^{\frac{5}{3}})/O(n\cdot k^{1.5})$$$. (thanks them.)

It is unimaginable to me that the solution of a seemingly simple problem is so difficult, Could any one find a better solution? Thank you !