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

A Combinatorial Approach to the Groebner Bases for Ideals Generated by Elementary Symmetric Functions