Extension of Newton-Girard to matrix-indexed monomials

Question

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 $$ E_d(x):=\sum_{1\leq a_1<\cdots< a_d \leq A} \sum_{1\leq b_1 < \cdots< b_d \leq B} x_{a_1,b_1} x_{a_2,b_2} \cdots x_{a_d,b_d} $$

Answer 1

Here is a recursive evaluation, that should be fast. Change of notation: $F_d[A,B]:=E_d(x)$. By recursing over the final term, $$ F_d [A,B] = x_{A,B}\cdot F_{d-1}[A-1,B-1] + \sum_{a_d\leq A-1} x_{a_d,B}\cdot F_{d-1}[a_d -1,B-1]+\\ \sum_{b_d\leq B-1} x_{A,b_d} F_{d-1}[A-1,b_d -1] + F_d [A-1,B-1] $$