The following question

$\quad$Problem about expectation of maximum partial sum:

can be reduced to determing

$$f(m,n) = \sum \max_{1\le i \le n}(x_1 + ... + x_i - \frac{m}{n} i)$$
where the sum is taken over all tuples $(x_1,...,x_n)$ of positive integers such that $x_1 + ... + x_n=m$.

Perhaps it's possible to evaluate this expression.