Round-Robin Tournaments and Forests

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?

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1 Answer

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.

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The polytope of win vectors, by J. E. Bartels, J. Mount and D. J. A. Welsh, is also accessible at springerlink.com/content/wr13217220x05726/fulltext.pdf and also speaks to this. –  ABh Aug 24 '10 at 2:21
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