TITLE: Problem about expectation of maximum partial sum
QUESTION [7 upvotes]: Given a number $m$, a random composition (strong) of this number into $n$ positive parts so that we can get $n$ random variable $X_1, X_2,\dots, X_n$ with $$X_1+X_2+\cdots+X_n=m$$ 
Note that all compositions of $m$ have the same probability.
Let $Y_i = X_i - \mathbb{E}(X_i)$, $S_i = Y_1 + \cdots + Y_i$
I want to calculate $\mathbb{E}(\max_{1 \le i \le n} S_i)$.

REPLY [2 votes]: I can't comment , but based on Noam D. Elkies's numbers and OEIS A000435 it looks as if 
$$c_n = \frac{(n-1)!}{2 n^n} \sum_{k=0}^{n-2} \frac{n^k}{k!}.$$