c_k q^k$, is the number of dimension $k$ operators in the basis. In the case of a conformal theory, the operator-state correspondence implies that the Hilbert series when only including EOM coincides with the partition function when we grade operators by their scaling dimension, angular momentum, and other quantum numbers.[^1]

Hilbert series are common objects in algebraic geometry and commutative algebra. They have appeared in the physics literature through their connection to invariant theory,[^2] finding application in, *e.g.*, studying flavor invariants [@Jenkins:2009dy; @Hanany:2010vu] and counting gauge invariants in SUSY gauge theories [@Benvenuti:2006qr; @Feng:2007ur; @Gray:2008yu]. A recent work reviewed and emphasised their utility for counting operators that do not contain derivatives in phenomenological settings, such as in subsets of the Standard Model EFT [@Lehman:2015via]. The present work can be seen as a contribution towards extending the use of Hilbert series in general EFTs by including operators with derivatives. Explicitly, we show how to systematically deal with IBPs and EOM, both of which define equivalence relations:

1.  **IBPs:** Two operators are equivalent if they are related by a total derivative, $$\scO_a \sim \scO_b \quad \text{if} \quad \scO_a = \scO_b + \text{d} \scO_c,
    \label{eqn:equiv_IBP}$$ since $\int_{\scM} \text{d} \scO = 0$.[^3]

2.  **EOM:** Two 