The elementary symmetric polynomial of degree 3 in variables x₁,x₂,x₃,x₄:

The zeroth elementary symmetric polynomial is defined to be 1:

The 2×3 matrices with entries 0 or 1:

Select matrices whose column sums are 1,1,1 and whose row sums are 2,1:

You can also count how many such matrices there are by using SymmetricPolynomial. The generating function of 2×3 matrices whose row sums are 2,1 is given by:

The coefficient of x₁¹x₂¹x₃¹ counts how many of these matrices have column sums 1,1,1:

The k elementary symmetric polynomial is the sum of all monomials constructed from k-subsets of the variables:

The generating function for the symmetric polynomials in variables is given by :

Check:

The monic polynomial with roots has coefficients that are elementary symmetric polynomials of the :

The elementary symmetric polynomials e_k=SymmetricPolynomial[k,{x₁,…,x_n}] are related to the power sum polynomials through the Newton–Girard formulas [MathWorld]. Generate all the Newton–Girard formulas for :

Verify them:

The elementary symmetric polynomials can be defined in terms of the generalized Bell polynomial BellY:

Verify for the case of five variables: