Invariance of Permanent of Matrix It is well known that the determinant of a matrix is invariant under some operations such as taking its transpose or row and column operations. Are there similar operations which fix the permanent of a matrix?
 A: In general immanants (which include permanents as a special case) are invariant under conjugation by any permutation matrix.  
For an $n\times n$ matrix $M$, this follows from the definition of the immanant:
\begin{align}
\hbox{Imm}^{\lambda}(M)&=\sum_{\sigma\in S_n}\chi^{\lambda}(\sigma)P_{\sigma}
\left[M_{11}M_{22}\ldots M_{nn}\right]\, ,\tag{1}\\
&=\sum_{\sigma\in S_n}\chi^{\lambda}(\sigma)
\left[M_{1\sigma(1)}M_{2\sigma(2)}\ldots M_{n\sigma(n)}\right]\, ,
\end{align}
where $\chi^{\lambda}(\sigma)$ is the character of element $\sigma$ in the irrep $\lambda$.
Conjugation by a permutation simply shuffles the ordering of the terms in the sum of elements in (1), but since all elements in the same class have the same character the sum itself remains invariant.
Minc has studied the more general problem in what types of linear transformations preserve the permanent (and other properties) in

Henryk Minc (1976) The invariance of elementary symmetric functions, Linear
  and Multilinear Algebra, 4:3, 209-215

A: The transpose does. Just stare at the formula. 
