The number of score sequences in tournaments

Abstract

In the te rminology of [1], a tou rnament is an asymmetr ic complete digraph, that is, a complete graph, each of whose ('~) lines is directed. The score of a poin t o f a t ou rnamen t is the number of lines directed away f rom it, and the score sequence is the ordered sequence of integers ( s l , s~ ..... s,,) compris ing all the scores of the points o f the tou rnamen t in non-decreasing order, that is, 0 ~< s~ ~< s2 ' ~< s,,. In [2], H. G. Landau has shown tha t a sequence (s~ ,..., s,,) is a score sequence if and only if S I ~ O,