a total derivative. For example, $(\pd^4\ph)\ph^{r-1}$ and $(\pd^3\ph)(\pd\ph)\ph^{r-2}$ are related via $\pd((\pd^3\ph)\ph^{r-1}) =0$.

The $p(k-1,r)$ relations obtained in the above manner are all independent[^6] and therefore the number of independent of operators modulo integration by parts is[^7] $$c_{k r, \text{IBP}} = p(k;r) - p(k-1;r),$$ and hence the Hilbert series is given by $$H_{1,\text{IBP}}(t,u) = \frac{1}{(u;t)_{\infty}} - \frac{t}{(u;t)_{\infty}} + t.$$ In the expansion of the second term above, the $t$ in the numerator ensures that the coefficient of $u^rt^k$ is $p(k-1;r)$. The sole $t$ in the above cancels the linear $u^0t^1$ piece in the expansion of the second term. The straightforward generalization to $N$ flavors is, $$H_{N,\text{IBP}}(t,\ui) = \frac{1-t}{\prod_{i=1}^N(u_i;t)_{\infty}} + t.
\label{eqn:H_IBP}$$

This result is also readily obtained using the formalism from section \[sec:framework\]. Let us consider the general $N$ flavor case. For fixed $r_1,\dots,r_N$, equivalence classes of operators related by integration by parts lie in the quotient ring $$\bigslant{R_{\br}^{G_{\br}}}{I_{\br}} = \bigslant{\mathbb{R}[\{P^{(1)}\},\dots,\{P^{(N)}\}]}{\avg{P_1^{(1)} + \dots + P_1^{(N)}}}.$$ The ideal reflects the statement of momentum conservation and accounts for integration by parts. In essence, this ideal removes one of the $P_1^{(i)}$ generators from $R_{\br}^{G_{\br}}$ when we construct the quotient ring. More precisely, when the ideal is defined by a single, homogeneous polynomial of 