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 $n(n+1)/4$.

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.