46.23 Permanent

Permanent( mat )

Permanent returns the permanent of the matrix mat. The permanent is defined by sum_{p in Symm(n)}{prod_{i=1}^{n}{mat[i][i^p]}}.

Note the similarity of the definition of the permanent to the definition of the determinant. In fact the only difference is the missing sign of the permutation. However the permanent is quite unlike the determinant, for example it is not multilinear or alternating. It has however important combinatorical properties.

    gap> Permanent( [[0,1,1,1],
    >                [1,0,1,1],
    >                [1,1,0,1],
    >                [1,1,1,0]] );
    9    # inefficient way to compute 'NrDerangements([1..4])'
    gap> Permanent( [[1,1,0,1,0,0,0],
    >                [0,1,1,0,1,0,0],
    >                [0,0,1,1,0,1,0],
    >                [0,0,0,1,1,0,1],
    >                [1,0,0,0,1,1,0],
    >                [0,1,0,0,0,1,1],
    >                [1,0,1,0,0,0,1]] );
    24    # 24 permutations fit the projective plane of order 2 

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