多校9 1001 Rikka with Nash Equilibrium

Nash Equilibrium is an important concept in game theory.

Rikka and Yuta are playing a simple matrix game. At the beginning of the game, Rikka shows an $$$n\times m$$$ integer matrix A. And then Yuta needs to choose an integer in $$$[1,n]$$$, Rikka needs to choose an integer in $$$[1,m]$$$. Let $$$i$$$ be Yuta's number and $$$j$$$ be Rikka's number, the final score of the game is $$$A_{i,j}$$$.

In the remaining part of this statement, we use $$$(i,j)$$$ to denote the strategy of Yuta and Rikka.

For example, when $$$n=m=3$$$ and matrix $$$A$$$ is
$$$$$$ \begin{bmatrix} 1 & 1 & 1 \\ 2 & 4 & 1 \\ 1 & 3 & 1 \\ \end{bmatrix} $$$$$$
If the strategy is $$$(1,2)$$$, the score will be $$$2$$$; if the strategy is $$$(2,2)$$$, the score will be $$$4$$$.

A pure strategy Nash equilibrium of this game is a strategy $$$(x,y)$$$ which satisfies neither Rikka nor Yuta can make the score higher by changing his(her) strategy unilaterally. Formally, $$$(x,y)$$$ is a Nash equilibrium if and only if:
$$$$$$\begin{cases} A_{x,y}≥A_{i,y}, & {\forall i \in [1,n]} \\[1ex] A_{x,y}≥A_{x,j}, & {\forall j \in [1,m]} \end{cases}$$$$$$
In the previous example, there are two pure strategy Nash equilibriums: $$$(3,1)$$$ and $$$(2,2)$$$.

To make the game more interesting, Rikka wants to construct a matrix $$$A$$$ for this game which satisfies the following conditions:
1. Each integer in $$$[1,nm]$$$ occurs exactly once in $$$A$$$.
2. The game has at most one pure strategy Nash equilibriums.

Now, Rikka wants you to count the number of matrixes with size $$$n×m$$$ which satisfy the conditions.