Function invariant under cyclic permutation of arguments

Question

I want to know if there exists a function invariant under cyclic permutation of arguments, but NOT invariant under any other permutation of arguments. For instance $$f(x,y,z) = x+y+z$$ is invariant under cyclic permutation of arguments $$f(x,y,z)=f(z, x,y) = f(y,z,x),$$ but is also invariant under any permutation of its arguments $$f(x,y,z)=f(x,z,y)=\ldots$$ The same happens for any function defined with a repeated application of a commutative operation. I want to know an example of a function that is invariant under cyclic permutation of arguments but NOT invariant under any other type of permutation. Thanks

Answer 1

$f(x,y,z) = x^2y + y^2z +z^2x$

$f(z,x,y) = z^2x + x^2y + y^2z=f(x,y,z)$

$f(y,z,x) = y^2z + z^2x + x^2y=f(x,y,z)$

However, $f(x,z,y)=x^2z+z^2y+y^2x \ne f(x,y,z)$

Answer 2

$$f(x,y,z) =x^4y^2z+y^4z^2x+z^4x^2y$$