  We give a combinatorial interpretation of the determinant of a matrix as a
generating function over Brauer diagrams in two different but related ways. The
sign of a permutation associated to its number of inversions in the Leibniz
formula for the determinant is replaced by the number of crossings in the
Brauer diagram. This interpretation naturally explains why the determinant of
an even antisymmetric matrix is the square of a Pfaffian.
