number of independent operators which contain $r_i$ powers of $\ph_i$ fields and $k$ derivatives, *i.e.* operators schematically of the form $\ph_1^{r_1}\dots\ph_N^{r_N}\pd^k$ where it is to be understood that the $k$ derivatives act in some unspecified way on the $\ph_1^{r_1}\dots\ph_N^{r_N}$.

The approach we take to computing $H_N$ is as follows. We fix the number of $\phi$ fields (*i.e.*, fix $\br$) and then count the number of derivatives we can add to form independent operators, that is, we consider the sum over $k$ in eq.  first. For fixed $\br$ this is also a Hilbert series, which we denote by $H_\br(t)=\sum_{k=0}^{\infty} c_{k\,\br}t^k $ (where $N$ is left implicit). The full Hilbert series is regained by the sum over all $H_\br(t)$ weighted by $\bu^\br$: $$H_N(t,\ui)=\sum_{\br=0}^{\infty} \bu^\br H_\br(t).$$ A key advantage in fixing $\br$ is that IBP only relates operators of the same $\br$. Moreover, for fixed $\br$ it makes sense to pass to Fourier space where implementing the equivalence under IBP and EOM becomes transparent.

We start by considering a single flavor and will then generalize the result to $N$ flavors. We perform a Fourier decomposition for the field $\ph$, writing $$\ph(\th) = \sum_{q=-\infty}^{\infty}a_qe^{iq\th},$$ where the $a_q$ are the Fourier coefficients. An operator composed 