point made at the end of section \[sec:framework\]: in order to make use of sum manipulations when computing the full Hilbert series, it is frequently simpler to leave $H_{\br}$ as a sum formula—as in eq. —rather than first finding a closed form sum for $H_{\br}$—as in eq. . While this distinction is mild for the present case, it is quite useful for the sums considered in the next subsection.

Relations from both integration by parts and equations of motion {#relations-from-both-integration-by-parts-and-equations-of-motion .unnumbered}
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We now turn to the task of computing the Hilbert series when we account for relations from both integration by parts and the equations of motion. As we will see, the Hilbert series in this case is much more interesting than when these relations are considered independently. Unlike the three previous cases, the generators of the EFT basis are not so easy to guess and there are non-trivial relations among them, leading to a rich structure in the Hilbert series.

Our launching point is the quotient ring of eq. , reproduced here for convenience, which describes equivalence classes of operators at fixed $\br$ $$\bigslant{R_{\br}^{G_{\br}}}{I_{\br}} = \bigslant{\mathbb{R}[P_1^{(1)},\dots,P_1^{(N)}]}{\big\langle P_1^{(1)}+\dots + P_1^{(N)},\big(P_1^{(1)}\big)^{r_1+1},\dots,\big(P_1^{(N)}\big)^{r_N+1}}\big\rangle.$$ We wish to find a sum formula for the 