Calculating the Frequency of Tournament Score Sequences

We indicate how to calculate the number of round-robin tournaments realizing a given score sequence. This is obtained by inductively calculating the number of tournaments realizing a score function. Tables up to 18 participants are obtained. 1. Tournaments and score sequences A (round-robin) tournament on a set P of n vertices (participants, teams, . . . ) is a directed graph obtained by orienting the complete graph Kn on P . In other words, a tournament is a directed graph on the vertex set P having exactly one arc connecting each pair in P . Clearly there are ( n 2 ) = n(n−1) 2 pairs in P to be connected by one of two possible arcs, and thus the total number of tournaments of size n is 2( n 2). The score function ft of a tournament t on P gives for each p ∈ P the outdegree ft(p) of p, i.e. ft(p) is the number of arcs of t leaving p. When the values of a score function ft are ordered (nondecreasingly, by convention) we obtain a score sequence. We say that this score sequence is realized by the tournament t. Three questions immediately arise concerning score sequences: (1) Which sequences are the score sequence of some tournament ? (2) How many different score sequences exist ? (3) How many tournaments realize a given score sequence ? The first question was solved by Landau [7] when investigating dominance relations within animal societies by the following result. Theorem 1.1 (Landau). Let s = (s1, s2, . . . , sn) be a nondecreasing sequence of nonnegative integers. Then s is the score sequence of some tournament if and only if