The permanent of a square matrix is known to have the following properties -
The permanent of the identity matrix is one.
For any permutation matrices $P$, $Q$, and matrix $A$, the permanent of $A$ equals the permanent of $PAQ$.
Is the permanent the only function of a square matrix that has these two properties? If not, what are some other examples?
This is definitely not true. Let $p(x_{11}, x_{12}, \cdots, x_{nn})$ be any symmetric polynomial of the variables such that $p(I)=1$ for the identity matrix (by abusing the notation, $I$ is identified with its entries), then $p$ satisfies both conditions: $PAQ$ is simply the result of switching rows and columns of $A$, so no entries have been changed. Further condtions are needed to characterize permanent.