Counting is Fun 101

You are given an array $A$ of $N$ elements, such that $1 \le A_i \le N$ for all $1 \leq i \leq N$. Count the number of permutation$^{\dag}$ pairs $(P, Q)$ where: - $P$ and $Q$ are permutations of length $N$; - $A_i = \max(P_i, Q_i)$ for all $1 \le i \le N$. Since the number can be huge, print it modulo $10^9 + 7$. $^{\dag}$A permutation of length $N$ consists of all integers from $1$ to $N$ exa