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When reading about the Coupon collector’s problem (CCP), I just came up with the following problem which I found very curious about, though it seems to have neither relation to the CCP nor practical value.

First, denote $S^{(k)}_n$ as a Stirling number of the second kind, a.k.a a number of ways to partition a set of $n$ labelled objects into $k$ nonempty unlabelled subsets. (This is the definition from the Wikipedia article:

The problem: Let $n$ be a fixed positive integer. Find the value of the positive integer $x$ such that the function $$f(n,x) = \frac{S^{(n-1)}_{x-1}}{n^x}$$ reaches its maximum.

Note: In fact, $n! f(n,x)$ is the probability of that the number of trials needed to collect all of the $n$ different coupons is $x$.

I haven’t had any ideas to solve the problem so far. (Actually, I had one, which was so naive and impractical) However, after several trials, I guess the function with the fixed $n$ will increase, reach its maximum and decrease to asymptotically zero while $x$ increases in the range of $\mathbb{Z^+}$.

Anyone has any ideas to solve the problem, please share. Thank you.

P.S: Sorry for my bad English.

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