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Number of distinct $f(x_1,x_2,x_3,\ldots,x_n)$ under permutation of the input

$\alpha _n ^n-1=0$ $\alpha _n=e^{2 \pi i/n}$ $$f(x_1,x_2,x_3,\ldots,x_n)=(x_1+\alpha _n x_2+ \alpha _n ^2 x_3+\cdots+\alpha _n ^{n-1} x_n)^n$$ I have read in Jim Brown's paper on page 5 that ...