Local Maxima

Given an $n \times m$ integer matrix $A$, a local maximum of $A$ is a location $(i, j)$ with $1 \le i \le n$ and $1 \le j \le m$ such that $A_{i, j}$ is no smaller than any other integer on the $i$-th row or on the $j$-th column.

For example, in the $3 \times 3$ matrix $\begin{bmatrix} 2 & 5 & 4 \\ 2 & 1 & 6 \\ 2 & 2 & 2 \end{bmatrix}\text{,}$ there are three local maxima: locations $(1, 2)$, $(2, 3)$, and $(3, 1)$ with values $5$, $6$, and $2$, respectively.

An $n \times m$ integer matrix $A$ is good if and only if it satisfies the following two conditions:

• There is exactly one local maximum in $A$.
• Each integer from $1$ to $n \times m$ occurs exactly once in $A$.

Given $n$, $m$, and a prime number $P$, count the number of good matrices of size $n \times m$ modulo $P$.

The first line contains three integers, $n$, $m$, and $P$, where $1 \le n, m \le 3000$ and $10^8 \le P \le 10^9 + 7$. It is guaranteed that $P$ is prime.

Output a single line with a single integer: the number of good matrices modulo $P$.