2019 USAJMO Problems/Problem 5

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:

1. for all $0\leq i,j\leq n$ , the set $S_{i,j}$ has $i+j$ elements; and 2. $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$ .

Proposed by Ricky Liu

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2019 USAJMO Problems/Problem 5