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](https://oeis.org/A001858) in the [OEIS](https://oeis.org/). But I can't see the correspondence between tournament standings and forests with labeled nodes. Can anyone explain this?



Round-Robin Tournaments and Forests
The bijection between score vectors and forests on labeled nodes is due to Kleitman and Winston. ([This](https://doi.org/10.1007/BF02579176 "Kleitman, D.J., Winston, K.J. Forests and score vectors. Combinatorica 1, 49–54 (1981). zbMATH review at https://zbmath.org/?q=an:0491.05028") paper)


A small clarification, your question about the cardinalities being equal was answered by Stanley and Zaslavsky, see Stanley's paper "[Decompositions of rational convex polytopes](https://doi.org/10.1016/S0167-5060(08)70717-9 "Stanley, R.P. Decompositions of rational convex polytopes. Ann. Discrete Math. 6, 333–342 (1980). zbMATH review at https://zbmath.org/?q=an:0812.52012")", but the proof was not bijective.


