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Math Help - Number of antisymmetric relationships

  1. #1
    Junior Member
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    Number of antisymmetric relationships

    Let A = {1,2,..,n}.
    What is the number of antisymmetric relationships on the set A?

    The answer is apparently 2^n * 3^((n^2-n)/2), but I don't know the proof.

    Thanks
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  2. #2
    MHF Contributor
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    If you represent an antisymmetric relation as a matrix, this matrix, call it A, can have either 0 or 1 on the diagonal. Let A[i, j] denote the element of A in row i and column j. If i <> j, then one or both of A[i, j] and A[j, i] must be 0, so there are three possibilities for the pair of locations (i, j) and (j, i).
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