Let $p$ be a prime number and $\mathbf{W}\subseteq \mathbb{F}_p[x]$ be the smallest set satisfying the following :
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(a) $x+1\in \mathbf{W}$ and $x^{p-2}+x^{p-3}+\cdots +x^2+2x+1\in \mathbf{W}$
(b) For $\gamma_1,\gamma_2$ in $\mathbf{W}$, we also have $\gamma(x)\in \mathbf{W}$, where $\gamma(x)$ is the remainder $(\gamma_1\circ \gamma_2)(x)\pmod {x^p-x}$.[/list]
How many polynomials are in $\mathbf{W}?$
1. **Define the set $\mathbf{W}$:**
   - $\mathbf{W}$ is the smallest set of polynomials in $\mathbb{F}_p[x]$ satisfying:
     - (a) $x+1 \in \mathbf{W}$ and $x^{p-2} + x^{p-3} + \cdots + x^2 + 2x + 1 \in \mathbf{W}$.
     - (b) For $\gamma_1, \gamma_2 \in \mathbf{W}$, the polynomial $\gamma(x)$, which is the remainder of $(\gamma_1 \circ \gamma_2)(x) \pmod{x^p - x}$, is also in $\mathbf{W}$.

2. **Characterize "nice" polynomials:**
   - A polynomial $f \in \mathbb{F}_p[x]$ is called *nice* if $f(0), f(1), \dots, f(p-1)$ are pairwise different.
   - Each nice polynomial can be associated with a permutation in $S_p$ (the symmetric group on $p$ elements).

3. **Lagrange Interpolation:**
   - By Lagrange Interpolation, for any permutation $\pi \in S_p$, there exists a polynomial $f \in \mathbb{F}_p[x]$ such that $f(i) = \pi(i)$ for $i = 0, 1, \dots, p-1$.
   - Any polynomial $g(x) = f(x) + (x^p - x)K(x)$, where $K(x) \in \mathbb{F}_p[x]$, will have the same values as $f$ at $0, 1, \dots, p-1$.

4. **Interpreting polynomials as permutations:**
   - The polynomial $x+1$ corresponds to the permutation $\alpha = (0 \ 1 \ 2 \ \cdots \ (p-1))$.
   - The polynomial $x^{p-2} + x^{p-3} + \cdots + x^2 + 2x + 1$ corresponds to the permutation $\beta = (0 \ 1 \ 2 \ \cdots \ (p-1))$.

5. **Closure under composition:**
   - If $\gamma_1, \gamma_2 \in \mathbf{W}$, then $\gamma_1 \circ \gamma_2 \in \mathbf{W}$ modulo $x^p - x$.
   - This implies that $\mathbf{W}$ is closed under composition of the corresponding permutations.

6. **Generating all permutations:**
   - The permutation $\alpha$ rotates all elements, and $\beta$ switches neighboring terms.
   - Using $\alpha$ and $\beta$, we can generate any transposition (swap of two elements).
   - Since transpositions generate the symmetric group $S_p$, we have $S_p \subseteq \mathbf{W}$.

7. **Conclusion:**
   - Since $\mathbf{W}$ contains all permutations and is a subset of $S_p$, we have $\mathbf{W} = S_p$.
   - The number of polynomials in $\mathbf{W}$ is $|S_p| = p!$.

The final answer is $\boxed{p!}$