The Newton-Girard recursions evidently give a fast algorithm for computing the elementary symmetric polynomials $e_d(x_1,\ldots,x_n):=\sum_{1\leq k_1<\cdots <k_d\leq n} x_{k_1}\cdots x_{k_d}$ in terms of power polynomials $p_d(x_1,\ldots,x_n) :=\sum_{k=1}^n x_k^d$. I am looking for an extension to summations over monomials indexed by a matrix, where the index ordering is enforced separately over each dimension.
Let $x:=\{x_{a,b}: a=1,\ldots A; b = 1,\ldots B\}$ be a two-dimensional array of variables. I want an extension of Newton-Girard (or some other fast recursion) for computing $$ E_d(x):=\sum_{1\leq a_1<\cdots< a_d \leq A} \sum_{1\leq b_1 < \cdots< b_d \leq B} x_{a_1,b_1} x_{a_2,b_2} \cdots x_{a_d,b_d} $$