Algebraic and combinatorial results for ranking competitors in a sequence of races

AbstractSuppose n competitors each compete in r races and a ranking function F assigns a score F(j) to the competitor finishing in the jth position in each race. The sum of the scores over the r races gives each competitor a final ranking with equal rankings being possible. A series representation and an asymptotic estimate are obtained for an, the number of ways of ranking n competitors in order, given that equal rankings are permissible. Also algebraic results are obtained which give criteria for the construction of a ranking function F which ranks scores in a predetermined way