  In an earlier work we described Gr\"obner bases of the ideal of polynomials
over a field, which vanish on the set of characteristic vectors $\mathbf{v} \in
\{0,1\}^n$ of the complete $d$ unifom set family over the ground set $[n]$. In
particular, it turns out that the standard monomials of the above ideal are
{\em ballot monomials}. We give here a partial extension of the latter fact. We
prove that the lexicographic standard monomials for linear Sperner systems are
also ballot monomials. A set family is a linear Sperner system if the
characteristic vectors satisfy a linear equation $a_1v_1+\cdots +a_nv_n=k$,
where $0<a_q\leq a_2\leq \cdots \leq a_n$ and $k$ are integers. As an
application, we confirm a conjecture of Frankl for linear Sperner systems.
