= Solution
The relation preserves the equality pattern among positions. The identity permutation gives reflexivity, inverses give symmetry, and compositions give transitivity.
For $W_3$, representatives are the restricted-growth words
$$
111;\qquad 111,112,121,122;\qquad 111,112,121,122,123
$$
when $n=1$, $n=2$, and $n\geq3$, respectively. Here digits denote distinct symbols, and each displayed word represents one class.
For $W_4$ with $n=3$, the fourteen classes have representatives
$$
1111,1112,1121,1122,1123,1211,1212,1213,
1221,1222,1223,1231,1232,1233.
$$
The cyclic <subgroup> $F$ gives finer classes than all of $P_4$. For example, $x_1x_1x_2$ and $x_1x_1x_3$ have the same equality pattern and are equivalent under $P_4$, but no power of the four-cycle fixes $x_1$ while sending $x_2$ to $x_3$. Thus the two equivalence-class decompositions of $W_3$ differ.
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