Antisimmetric relations question

[doktorwho]

Homework Statement

A set $P=\left\{ \ p1,p2,p3,p4 \right\}$ is given. Determine the number of antisimmetric relations of this set so that $p1$ is in relation with $p3$, $p2$ is in relation with $p4$ but $p2$ is not in relation with $p1$.

Homework Equations

3. The Attempt at a Solution [/B]
By drawing the table of relations i concluded that the total number of antisimmetrical relations without any constriction further implied in the problem is $2^n3^{\frac{n^2-n}{2}}$ with the first factor drawn from the diagonal possibilities and the second factpr being 3 possibilities on the upper part above the diagonal. I do not however know how to include these steps. Can you help?

 

[nuuskur]

This is the picture as I understand it.
<br /> \begin{array}{ccc}x&amp;x&amp;1&amp;x\\0&amp;x&amp;x&amp;1\\x&amp;x&amp;x&amp;x\\x&amp;x&amp;x&amp;x\end{array}<br />
You correctly deduce that there are a total of $2^n\cdot 3^{\binom{n}{2}}$ anti-symmetrical relations on a set of $n$ elements. There's nothing special to this, you have fixed 2 components in the upper triangle bit. Keep in mind what an anti-symmetrical relation matrix has to look like and use the product rule.