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?



Which of these sums appear most often?
In fact this question was already asked at MO, although in disguise: see [here](https://mathoverflow.net/questions/58329/partitions-to-different-parts-not-exceeding-n). 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](http://math.mit.edu/~rstan/algcomb/algcomb.pdf).


