r$ is unnecessary—in this situation $H_{\br}(R_{\br}^{G_{\br}}/I_\br,t)$ is relatively simple to compute. If $r_j < \sum_{i\ne j}r_i$, then it is necessary to include $I_{2k \le r}$ while the conditions $\{k\le r - r_i\}$ are automatically satisfied.

With eq. , we can obtain the closed form of the Hilbert series for relatively low numbers of flavors, $N \le 3$ $$\begin{aligned}
H_1 &=& \frac{1}{{1 - {u_1}}} , \nonumber\\
H_2 &=& \frac{1}{{\left( {1 - {u_1}} \right)\left( {1 - {u_2}} \right)\left( {1 - t{u_1}{u_2}} \right)}} , \label{eq:someres} \\
H_3 &=& \frac{{1 - t{u_1}{u_2}{u_3}}}{{\left( {1 - {u_1}} \right)\left( {1 - {u_2}} \right)\left( {1 - {u_3}} \right)\left( {1 - t{u_1}{u_2}} \right)\left( {1 - t{u_1}{u_3}} \right)\left( {1 - t{u_2}{u_3}} \right)}} \nonumber.\end{aligned}$$ For larger $N$, it becomes very laborious to directly sum up the expression eq. . Moreover, the form of the Hilbert series becomes increasingly complicated. For example, for $N=4,5$ we have H\_4 &=& ,\
H\_5 &=& , where $s_m$ are the elementary symmetric polynomials in $N$ variables $u_1,\dots,u_N$, $$s_m = \sum_{1\le i_1<\dots<i_m \le N}u_{i_1}\dots u_{i_m},$$ (and where the value of $N$ for $s_m$ is left implicit in eqs. ). These can be more readily obtained via residues that the sum formula eq.  can be shown to have 