sumsS

SumsS is a term used in some mathematical texts to denote the family of symmetric sums associated with a finite multiset of variables. It collects all elementary symmetric sums of the multiset, organized by the size of the contributing subsets. Given variables x1, x2, ..., xn, sumsS consists of s_k = sum over all 1 ≤ i1 < ... < ik ≤ n of x_{i1} x_{i2} ... x_{ik} for k = 0, 1, ..., n, with s_0 = 1. The collection sumsS is the sequence {s_0, s_1, ..., s_n}.

The generating function for sumsS is the formal power series sum_{k=0}^n s_k t^k = ∏_{i=1}^n (1 + x_i t). Equivalently, the monic polynomial with roots -x_i can be written as ∏_{i=1}^n (t - x_i) = t^n − s_1 t^{n−1} + s_2 t^{n−2} − ... + (−1)^n s_n. This relation is a form of Viète’s formulas, linking sumsS to the coefficients of polynomials with given roots.

Computation is typically done by an iterative or dynamic-programming approach: incorporating each variable x_i updates the current sumsS via s_k^{(i)} = s_k^{(i−1)} + x_i s_{k−1}^{(i−1)}, with base cases s_0^{(0)} = 1 and s_k^{(0)} = 0 for k > 0. This yields an O(n^2) time algorithm. Generalizations of sumsS extend to multisets, to elements of arbitrary commutative rings, or to weighted variants, with similar generating-function structures.

Applications of sumsS include polynomial factorization, solving equations via Viète’s formulas, and generating-function techniques in combinatorics. Example: for x = {2, 3, 5}, one obtains s_0 = 1, s_1 = 10, s_2 = 31, s_3 = 30. See also: elementary symmetric polynomials, Viète’s formulas, generating functions, Newton’s identities.