Is there an equation in $x$, $y$ and $z$ where there are exactly $2$ permutations in the variables that give fundamentally different polynomials? For example:
$$x+y+z$$ gives only one permutation $$x+y+z^2$$ gives 3 permutations (we basically get to pick which variable replaces $z$) $$x+y^2+z^3$$ gives 6 permutations (essentially $S_3$).
I've read that there is one for each factor of $3!$, but I don't see one for $2$. The examples above cover the obvious cases: all the same, two the same and all different. So is there an example with 2 permutations?
Try... $$x y^2 + y z^2 + z x^2$$