Which of these sums appear most often?

Question

Let $N=\{1,2,3,\ldots, n\}$.
We sum all the elements of every nonempty subset of $N$.
Which sum(s) appears most often? (Let's call this sum a champion).
Using a simple pigeonhole argument a champion must appear at least $\frac{2^n-1}{T_n}$ times. ($T_n$ denotes the $n$-th triangular number).
It seems that the champion should be somewhere around $T_n/2$ but I cannot prove it.

Am I missing something obvious here?

Answer 1

In fact this question was already asked at MO, although in disguise: see here. Richard Stanley answered it wonderfully. The champions are the nearest integers to $n(n+1)/4$.

For a quick proof, see Lemma 6.13 on Page 93 (and the preliminaries on Page 92) in Stanley's Topics in algebraic combinatorics.