{T}he Gr\"obner Basis of a Catalan Path Ideal

For the ideal $I=⟨y_1+\cdots +y_n,y_1^2,\dots ,y_n^2⟩$ in $R=\mathbb{F}[y_1,\dots ,y_n]$ with char($\mathbb{F}$) = 0, we show that the reduced Gr"obner basis with lex-order consists of polynomials $g_{\alpha }$ that are represented in terms of paths, moving northeast in the Cartesian plane, that stay above the diagonal and cross the diagonal at the last step. This implies that a linear basis for the quotient ring $R\mathrm{/}I$ is given by a set of Catalan paths. We show that the dimension is the number of standard Young tableaux of size $n$ and height at most two. The graded Frobenius characteristic of $R\mathrm{/}I$ as a symmetric group module is given by ${\sum }_{k=0}^{\lfloor \frac{n}{2}\rfloor }{s}_{n-k,k}{q}^k$.