Elementary symmetric polynomial

Definition
The elementary symmetric polynomial in $$m$$ variables, associated with a partition $$\lambda = (\lambda_1,\lambda_2,\ldots,\lambda_r)$$ is defined as the product of the elementary symmetric polynomials associated with each $$\lambda_i$$. Here, the elementary symmetric polynomial associated with a positive integer $$l$$ is the sum of all monomials in the $$m$$ variables having degree $$l$$.

Nature of the polynomials
The elementary symmetric polynomials are all symmetric polynomials. In fact, the elementary symmetric polynomials corresponding to all tableaux with $$n$$ cells, in $$m$$ variables, form a basis for the space of all homogeneous symmetric polynomials in $$m$$ variables, of degree $$n$$.

Complete symmetric polynomials
The map that changes basis from elementary symmetric polynomials to complete symmetric polynomials is in fact an involutive ring homomorphism. This follows from the fact that for any $$t$$, the product of the formal power series $$e(t)$$ and $$h(-t)$$ is 1, where:

$$e(t) = \sum_{k \ge 0} e_kt^k$$

and

$$h(t) = \sum_{k \ge 0} h_kt^k$$