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Reference for permanent integral identity

$\DeclareMathOperator\perm{perm}\DeclareMathOperator\diag{diag}$Using MacMahon's master theorem, the properties of complex gaussian integrals, and Cauchy's integral theorem one can show that the permanent … To prove it directly, we note that MacMahon's master theorem states that the permanent of a matrix $A$ is the $x_1x_2\dotsm x_N$ coefficient of the quantity \begin{equation} \frac{1}{\det(I - XA)}, \end …
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Reference for permanent integral identity

Then the permanent of $A$ is \begin{equation} \text{perm}(A) = \int_{\Omega^n_{X}} d^n\textbf{x}\, \prod_{i=1}^n p_{X}(x_i) \, x_{i} \sum_{j=1}^n a_{i, j} x_j, \label{eq:fund_thm} \end{equation} where …
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