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Antisimmetric relations question

  1. Apr 9, 2017 #1
    1. The problem statement, all variables and given/known data
    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##.
    2. Relevant equations
    3. The attempt at a solution

    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?
     
  2. jcsd
  3. Apr 9, 2017 #2
    This is the picture as I understand it.
    [tex]
    \begin{array}{ccc}x&x&1&x\\0&x&x&1\\x&x&x&x\\x&x&x&x\end{array}
    [/tex]
    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.
     
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