  This paper defines the Iris function and provides two formulations of the
matrix permanent. The first formulation, valid for arbitrary complex matrices,
expresses the permanent of a complex matrix as a contour integral of a second
order Iris function over the unit circle around zero. The second formulation is
defined for the restricted set of matrices with complex or "Gaussian" integer
elements. Using the second formulation, the paper shows that the computation of
the permanent of an arbitrary $n\times n$ 0-1 matrix is bounded by $o\left(
n^{27} \left( \log \left( {{n^3}} \right) \right)^6 \left( \log_2 \left( {{n}}
\right) \right)^2\right)$ binary operations.
