Question 1
For a positive integer $n$, an $n$-configuration is a family of sets $\langle A_{i,j} \rangle_{1\le i,j\le n}$. An $n$-configuration is called *sweet* if for every pair of indices $(i, j)$ with $1 \le i \le n-1$ and $1 \le j \le n$ we have $A_{i,j} \subseteq A_{i+1,j}$ and $A_{j,i} \subseteq A_{j,i+1}$. Let $f(n,k)$ denote the number of sweet $n$-configurations such that $A_{n,n} \subseteq \{1, 2, \dots, k\}$. Determine which number is larger: $f(2024, 2024^2)$ or $f(2024^2, 2024)$.