How many Antisymmetric relations are possible on a set A?

Question

Let $A$ be a set with $|A| =5$. How many Antisymmetric relations are possible on set A whose size is maximum ?

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As I started with small cases, the maximum size of antisymmetric relation that I got is $\frac{n(n+1)}{2}$ which gives maximum size = 15

But, I am having a hard time to find out how many are there ?

Answer 1

So given that it is maximal, it will have all pairs $(a,a)$ with $a \in A$.

And of all pairs of different objects $a$ and $b$ it either has $(a,b)$ or $(b,a)$ in it, but not both.

There are 10 such pairs, so that means there are $2^{10}$ such possible maximum anti-symmetric relations.