USAMO 2019 — Days 1 and 2 Question 4

Let $n\ge0$ be an integer. Count families of sets $S_{ij}\subseteq\{1,\ldots,2n\}$ for $0\le i,j\le n$ such that $|S_{ij}|=i+j$ and $S_{ij}\subseteq S_{kl}$ whenever $i\le k$ and $j\le l$. The sets need not be distinct.

Proposed by Ricky Liu. Source: Evan Chen’s third-party solution notes.