c_{k\, \br, \text{free}} \bu^{\br}t^k$. The number of operators composed of $k$ derivatives acting on $\br$ powers of $\ph$ fields is the number of ways of partitioning the $k$ derivatives onto $\ph_1^{r_1}\dots \ph_N^{r_N}$. Specifically, for $N=1$, $c_{k\, r, \text{free}} = p(k;r)$ is the number of (indistinct) partitions of $k$ into at most $r$ parts.[^5] For general $N$, $$c_{k r_1\dots r_N, \text{free}} = \sum_{k_1+\dots+k_N = k}p(k_1;r_1)\dots p(k_N;r_N),$$ *i.e.* $c_{k\, \br,\text{free}}$ is the number of ways of distinctly partitioning $k$ into $N$ parts ($k_1+ \dots + k_N = k$) weighted by the number of indistinct partitions of the $k_i$ into at most $r_i$ parts.

Only relations from integration by parts {#only-relations-from-integration-by-parts .unnumbered}
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Operators which are total derivatives vanish in the action, leading to relations from integration by parts. For simplicity, let us consider the one flavor case and count the number of operators composed of $r$ powers of $\ph$ fields and $k$ derivatives, modulo integration by parts. Per the discussion above, the number of operators of the form $\ph^r\pd^k$ is the number of partitions of $k$ into at most $r$ parts, $p(k;r)$. Relations amongst the $p(k;r)$ operators from IBP are formed from taking the $p(k-1;r)$ operators with one less derivative and applying 