MathDB

For a positive integer

n

, an

n

-configuration is a family of sets

\left\langle A_{i,j}\right\rangle_{1\le i,j\le n}.

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,\ldots , k\}

. Determine which number is larger:

f\left(2024, 2024^2\right)

f\left(2024^2, 2024\right).

2