Asymptotic Enumeration of Tournaments with a Given Score Sequence

A tournament is a digraph in which, for each pair of distinct vertices v and w, either (v,w) or (w, v) is an edge, but not both. A tournament is regular if the in-degree is equal to the out-degree at each vertex. Let v1, v2, . . . , vn be the vertices of a labelled tournament and let d−j , d + j be the in-degree and out-degree of vj for 1 ≤ j ≤ n. d+j is also called the score of vj . Define δj = d + j − d−j and call δ1, δ2, . . . , δn the excess sequence of the tournament. Let NT (n; δ1, . . . , δn) be the number of labelled tournaments with n vertices and excess sequence δ1, . . . , δn. It is clear that NT (n; δ1, . . . , δn) = 0 unless all the excesses have different parity from n; we will assume this without further mention for the entire paper. As in [3], let RT (n) = NT (n; 0, . . . , 0) be the number of labelled regular tournaments with n vertices. The first attack that we are aware of on the asymptotics of tournaments was due to Joel Spencer [6]. In particular, Spencer evaluated RT (n) to within a factor of (1 + o(1)) and obtained the estimate