Problem

Source: 2019 USAMO Problem 4/USAJMO Problem 5

Tags: AMC, USA(J)MO, USAMO, Olympiad Combinatorics, recursion, 2019 USAJMO, 2019 USAMO



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: for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$. Proposed by Ricky Liu