On Some Optimal Constructions of Chess Tournament Combinations

摘要

An optimal solution for the following “chess tournament” problem is given. Letn,rbe positive integers such thatr0 impliesj=k, tij=0 is equivalent toaij=0, and ai1+ai2+…+aiN=R (1⩽i⩽N). Letp(T, A) be the number ofisuch that 1⩽i⩽Nandai1+ai2+ … +aiN>0. The main result of this note is to show that maxp(T, A) for (T, A)∈XN, Ris equal to n(2r+1)/(r+1), and a pair (T0,A0) satisfyingp(T0,A0