On linear programming duality and Landau's characterization of tournament scores

H. G. Landau has characterized those integer-sequences S - (s(1), s(2),..., s(n)) which can arise as score-vectors in an ordinary roundrobin tournament among n contestants [17]. If s(1) <= s(2) <= center dot center dot center dot <= s(n), the relevant conditions are expressed simply by the inequalities: Sigma(k)(i=1) s(i) >= (k 2), for k = 1, 2,..., n, with equality holding when k = n. The necessity of these conditions is fairly obvious, and proofs of their sufficiency have been given using a variety of different methods [1, 2, 4, 10, 22, 23]. The purpose of this note is to exhibit Landau's theorem as an instance of the "duality principle" of linear programming, and to point out that this approach suggests an extension of Landau's result going beyond the well-known generalizations due to J.W. Moon [20, 19].