Computation of the Number of Score Sequences in Round-Robin Tournaments

Abstract

We consider round-robin tournaments of n players in which, at each encounter, the winner is awarded 1 point and the loser 0 (ties are excluded).Let1be the n scores, ordered in a non-decreasing sequence. Clearly2

A bernoulli excursion and its various applications

“…The continued fraction (22) was encountered by Ramanujan [24] in the theory of partitions. (See for instance Hardy and Wright [13], p.…”

“…In 1968 in studying the continued fraction(22) Szekeres[28] observed that the coefficients fn(n + 2j) in(23)have simple combinatorial interpretations. He proved that fn(n + 2j), where j =0, 1, • • • ,(~), can be interpreted as the number of positive integers at, a2, ... , an satisfying the conditions 1~at~a2~...~an,…”

Condition for a tournament score sequence to be simple

“…Let s(n) denote the number of simple score sequences of order n. It is easy to show that s ( n ) satisfies the following recurrence relation, which can be used to evaluate s(n): Table I lists for small values of n the simple score sequences of order n and compares s(n) with t(n), the total number of score sequences of order n, enumerated in [7].…”

Counting Various Classes of Tournament Score Sequences

“…This test was utilized by Bent [2] in a remarkable 1964 Master of Science dissertation. See also Narayana and Bent [8]. We present Bent's algorithm here because we will need it in later sections.…”