$f(x_1,x_2,x_3,\ldots,x_n)$ Maximum how many different results can have with all permutation of inputs?

$\alpha _n=e^{2 \pi i/n}$

$$f(x_1,x_2,x_3,\ldots,x_n)=(x_1+\alpha _n x_2+ \alpha _n ^2 x_3+\cdots+\alpha _n ^{n-1} x_n)^n$$

Maximum how many different results can have with all permutation of inputs? I have read in Jim Brown's paper on page 5. http://www.math.caltech.edu/~jimlb/abel.pdf

Lagrange showed that If n=3 then $f(x_1,x_2,x_3)$ Maximum can have 2 different results with all permutations of $(x_1,x_2,x_3)$

If n=4 then $f(x_1,x_2,x_3,x_4)$ Maximum can have 3 different results with all permutations of $(x_1,x_2,x_3,x_4)$

If n=5 then $f(x_1,x_2,x_3,x_4,x_5)$ Maximum can have 6 different results with all permutations of $(x_1,x_2,x_3,x_4,x_5)$

Is there any general formula for n and which method is used to find the general formula?

The function you describe involve what we call Lagrange resolvents. The situation is a bit more complicated. For an eqn of odd prime degree $p$, you need a resolvent of degree $p-1$ with coefficients that are algebraic numbers of $(p-2)!$ deg. Thus, for $p=3$, you only get a quadratic resolvent. However, for $p=5$, you'll need to solve a $(5-2)!=3!=6$ deg eqn. For $p=7$ it is already $(7-2)!=5!=120$ deg eqn. And so on. – Tito Piezas III Jul 8 at 2:25