  Previous work by Mora and Sala provides the reduced Groebner basis of the
ideal formed by the elementary symmetric polynomials in $n$ variables of
degrees $k=1,\dots,n$, $\langle e_{1,n}(x), \dots, e_{n,n}(x) \rangle$.
Haglund, Rhoades, and Shimonozo expand upon this, finding the reduced Groebner
basis of the ideal of elementary symmetric polynomials in $n$ variables of
degree $d$ for $d=n-k+1,\dots,n$ for $k\leq n$. In this paper, we further
generalize their findings by using symbolic computation and experimentation to
construct the reduced Groebner basis for the ideal generated by the elementary
symmetric polynomials in $n$ variables of arbitrary degrees.
