On the Wikipedia page for binary relations, there's a table that includes the number of distinct binary relations given an $n$-element set: $2^{n^2}$. They also include how many distinct equivalence relations, total orders, partial orders, etc.
I know the other ones only make sense in the context of binary relations but, how might we generalize this result and find the number of distinct $n$-ary relations given a set of $k$ elements?
A n-ary relation on a set K of k elements is a subset of K$^n$.
K$^n$ has k$^n$ elements.
Thus there is how many n-ary relations?