Suppose $\displaystyle f,g \in X $ and have the same pattern. Prove that $\displaystyle W(f) = W(g) $.

Now an orbit of and element $\displaystyle f $ in $\displaystyle X $ is all the "possible stuff" that a permutation $\displaystyle \pi $ can map $\displaystyle f $ to. So does $\displaystyle f,g \in X $ imply that they are in the same orbit?