In THE SLOPES DETERMINED BY n POINTS IN THE PLANE by JEREMY L. MARTIN, Page 2, Theorem 1.1, a Hilbert series is used to compute some combinatorial objects:

Let $R_n=k[m_{1,2}, \ldots, m_{n-1,n}]$, and let $I_n$ be the ideal generated by the tree polynomials of all rigidity circuits in the complete graph $K_n$. Then the coefficient $h(n,k)$ of the numerator of the Hilbert series of $R_n/I_n$ counts the number of perfect matchings on $[1, 2n − 4]$ with exactly $k$ long pairs, that is, pairs not of the form $\{i,i + 1\}$.

In general, if we want to compute some combinatorial objects using Hilbert series, how to define the ring $R_n$ and the ideal $I_n$?

Are there some other examples which count some combinatorial objects using Hilbert series?

Thank you very much.