Meaning of$~\sum_\text{cyclic}$?

There are six permutations of the variables $(x, y, z)$, namely $(x, y, z)$, $(x, z, y)$, $(y, x, z)$, $(y, z, x)$, $(z, x, y)$, and $(z, y, x)$.

There are three cyclic permutations of the variables $(x, y, z)$, namely $(x, y, z)$, $(y, z, x)$, and $(z, x, y)$.

So the sum of the expression $(x - y)^z$ over all permutations is given by

$$(x - y)^z + (x - z)^y + (y - x)^z + (y - z)^x + (z - x)^y + (z - y)^x$$

whereas the sum of the expression $(x - y)^z$ over all cyclic permutations is given by

$$(x - y)^z + (y - z)^x + (z - x)^y.$$

Note that the two sums are not the same; the first contains more summands as there are more permutations than cyclic permutations (i.e. not every permutation is a cyclic permutation, e.g. $(x, z, y)$ is a permutation of $(x, y, z)$ but not a cyclic permutation).