Say I have a matrix
\begin{align*} G &= \left( { \begin{array}{cc} G^{\prime} & \vec{u} \\ \vec{v}^T & d \end{array} } \right) \end{align*}
where $\vec{v}^T$ and $\vec{u}$ are row and column vectors, respectively, and $d$ is a diagonal entry. $\vec{v}$, $\vec{u}$, and $d$ are all given. Is there a closed-form expression for the permanent of $G$ in terms of the permanent of the remaining sub-matrix, $G^{\prime}$, the entries of $\vec{u}$ and $\vec{v}$, and $d$? The cofactor formula for the permanent seems encouraging, but I can't quite seem to make it fit.
No, there isn't. For example, consider $G' = \pmatrix{a_{11} & a_{12}\cr a_{21} & 0\cr}$, $\vec{u} = \pmatrix{a_{13}\cr a_{23}\cr}$, $\vec{v}^T = (a_{31}, a_{32})$ and $d = a_{33}$. Note that the permanent of $G'$ is $ a_{12} a_{21}$, which does not depend on $a_{11}$.
If the permanent of $G$ depended only on the permanent of $G'$ and the entries of $\vec{u}$, $\vec{v}$ and $\vec{d}$, it would not depend on $a_{11}$ either. But it does: it is $a_{{1,1}}a_{{2,3}}a_{{3,2}}+a_{{1,2}}a_{{2,1}}a_{{3,3}}+a_{{1,2}}a_{{2 ,3}}a_{{3,1}}+a_{{1,3}}a_{{2,1}}a_{{3,2}}$