Question

In a round-robin tournament with $n$ teams, each team plays every other team exactly once. Thus, there are $n(n-1)/2$ total games played. How many different standings can result? By a "standing" I mean the ordered sequence $(W_1, \ldots, W_n)$ where $W_i$ is the number of wins by the $i$th player. Assume that no game ends in a tie.

I wrote a program to calculate this sequence. For $n$ up to 13, it agrees with the number of forests on $n$ labeled nodes, which is Sequence A001858 in the OEIS. But I can't see the correspondence between tournament standings and forests with labeled nodes. Can anyone explain this?

Answer 1

The bijection between score vectors and forests on labeled nodes is due to Kleitman and Winston. (This paper)

A small clarification, your question about the cardinalities being equal was answered by Stanley and Zaslavsky, see Stanley's paper "Decomposition of rational convex polytopes", but the proof was not bijective.