Permanent/Examples/Matrix whose Entries are Product of Row and Column Indices

Example of Permanent
The square matrix of the form:
 * $\begin{pmatrix}

1 \times 1 & 1 \times 2 & \cdots & 1 \times m \\ 2 \times 1 & 2 \times 2 & \cdots & 2 \times m \\ \vdots & \vdots & \ddots & \vdots \\ m \times 1 & m \times 2 & \cdots & m \times m \end{pmatrix}$

has a permanent of $\left({n!}\right)^3$.

Proof
There are $n!$ terms in a permanent.

By its structure, each one of these has one element from each row multitplied by one element from each column.

Thus each term of the permanent consists of:
 * $\left({1 \times 2 \times \cdots \times n}\right) \times \left({1 \times 2 \times \cdots \times n}\right)$

in some order, that is:
 * $\left({n!}\right)^2$

As has been stated, there are $n!$ of these.

Thus the permanent of this matrix is $n! \left({n!}\right)^2$.

Hence the result.