Landau's inequalities for tournament scores and a short proof of a theorem on transitive sub-tournaments

Ao and Hanson, and Guiduli, Gy arf as, Thomass e and Weidl independently, proved the following result: For any tournament score sequence S = (s1; s2; : : : ; sn) with s1 s2 sn, there exists a tournament T on vertex set f1; 2; : : :; ng such that the score of each vertex i is si 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 satis es 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.