Landau ' s Inequalities for Tournament Scores anda

Ao and Hanson, and Guiduli, Gyy arff as, Thomass e and Weidl independently, proved the following result: For any tournament score sequence S = (s 1 ; s 2 ; : : : ; s n) with s 1 s 2 s n , there exists a tournament T on vertex set f1; 2; : : : ; ng such that the score of each vertex i is s i and the sub-tournaments of T on both the even and the odd indexed vertices are transitive in the given order; that is, i dominates j whenever i > j and i j (mod2). In this note, we give a much shorter proof of the result. In the course of doing so, we show that the score sequence of a tournament satisses a set of inequalities which are individually stronger than the well-known set of inequalities of Landau, but collectively the two sets of inequalities are equivalent.