We call a monic polynomial $P(x) \in \mathbb{Z}[x]$ [i]square-free mod n[/i] if there [u]dose not[/u] exist polynomials $Q(x),R(x) \in \mathbb{Z}[x]$ with $Q$ being non-constant and $P(x) \equiv Q(x)^2 R(x) \mod n$. Given a prime $p$ and integer $m \geq 2$. Find the number of monic [i]square-free mod p[/i] $P(x)$ with degree $m$ and coeeficients in $\{0,1,2,3,...,p-1\}$.

[i]Proposed by Masud Shafaie[/i]
**
   - For \( n \geq 2 \), we solve the recurrence relation. It can be shown that:
     \[
     |S_n| = p^n - p^{n-1}
     \]
   - This is derived by induction or by solving the recurrence relation directly.

The final answer is \( \boxed{ p^m - p^{m-1} } \) for \( m \geq 2 \).