Complete symmetric polynomials are a special class of polynomials that are formed from a set of variables. For a given number of variables, the complete symmetric polynomial of degree n is the sum of all possible products of n variables, allowing for repetitions. For example, in two variables x and y , the complete symmetric polynomials include terms like x^2, xy, and y^2 . These polynomials play a significant role in various areas of mathematics, including combinatorics and representation theory. They are closely related to symmetric functions and can be used to express other types of symmetric polynomials, such as power sums and elementary symmetric polynomials.