2011 IMO 第 4 题

Let $n > 0$ be an integer. We are given a balance and $n$ weights of weight $2^{0}$ , $2^{1}$ , ..., $2^{n - 1}$ . We are to place each of the $n$ weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed. Determine the number of ways in which this can be done.

设 $n > 0$ 为整数。我们得到一个余额和 $n$ 个权重，权重为 $2^{0}$ 、 $2^{1}$ 、...、 $2^{n - 1}$ 。我们将每个 $n$ 砝码一个接一个地放在天平上，确保右盘永远不会重于左盘。在每一步中，我们选择一个尚未放置在天平上的砝码，并将其放置在左盘或右盘上，直到所有砝码都放置完毕。确定可以实现此目的的方法数量。