2019 USAMO 第 4 题

Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that:

$\bullet$ for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and

$\bullet$ $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$.

令 $n$ 为非负整数。确定 $(n+1)^2$ 可以选择的方式数设置 $S_{i,j}\subseteq\{1,2,\ldots,2n\}$，对于整数 $i,j$ 和 $0\leq i,j\leq n$，这样：

$\bullet$ 对于所有 $0\leq i,j\leq n$，集合 $S_{i,j}$ 有 $i+j$ 个元素；和

$\bullet$ $S_{i,j}\subseteq S_{k,l}$ 每当 $0\leq i\leq k\leq n$ 和 $0\leq j\leq l\leq n$ 时。