I have the following polynomial in the variables $A_1, A_2, ..., A_n;B_1, ...,B_n:$
$ f(A_1, A_2, ..., A_n;B_1, ...,B_n) = A_1B_2B_3...B_n + B_1A_2B_3...B_n + ...+B_1...B_{n-1}A_n$
Without the $A_j$ I would get the elementary symmetric polynomial $e_{n - 1}(B_1, ...B_n)$, hence the question: Is there a way to rewrite f into a nicer form, maybe using symmetric polynomials, or generating functions?