Landau's theorem on tournament scores

A nondecreasing list of nonnegative integers $s_1\le \cdots \le s_n$ is the list of outdegrees of a tournament exactly whenNecessity follows because any $k$ vertices have $\left(\genfrac{}{}{0ex}{}{k}{2}\right)$ internal edges, each contributing one to their total outdegree. Sufficiency is the substantive part of Landau's theorem on tournament scores.

The list $(1,1,1)$ is realized by a directed triangle; $(0,1,2)$ is realized by a transitive tournament. The list $(0,0,3,3)$ has the correct total but fails the inequality at $k=2$, so it cannot occur. Total score alone therefore does not characterize a tournament.