MathDB

A tournament of order

n

n \in \mathbb{N}

, consists of

2^n

players, which are numbered with

1, 2, \ldots, 2^n

, and has

n

rounds. In each round, the remaining players paired with each other to play a match and the winner from each match advances to the next round. The winner of the

n

-th round is considered the winner of the tournament. Two tournaments are considered different if there is a match that took place in the

k

-th round of one tournament, but not in the

k

-th round of the other, or if the tournaments have different winners. Determine how many different tournaments of order

n

there are with the property that in each round, the sum of the numbers of the players in each match is the same (but not necessarily the same for all rounds).

Problem Statement