Let $n$ and $k$ be two given numbers. The goal is to choose $n$ subsets from $\{1,2,...,n\}$ such that the union of any $k$ of these subsets is the set $\{1,2,...,n\}$ and the union of any $m < k$ of these subsets is not the entire set $\{1,2,...,n\}$.

If $n$ and $k$ are given numbers, then how many ways are there to choose $n$ subsets from $\{1,2,...,n\}$ satisfying the given conditions?

For example, if $n=3$ and $k=2$, there is only one case $(\{1,2\} , \{2,3\} , \{3,1\})$ to choose three subsets from $\{1,2,3\}$ such that every two of these subsets construct $\{1,2,3\}$ and every one of these subsets do not construct the set $\{1,2,3\}$.

It is a kind of secret sharing in the cryptography that we want to distribute $n$ key between $n$ people such that at least we need $k$ people to decrypt.