  In this work, we explore an extension of Hilbert series techniques to count
operators that include derivatives. For sufficiently low-derivative operators,
we find an algorithm that gives the number of invariant operators, properly
accounting for redundancies due to the equations of motion and integration by
parts. Specifically, the technique can be applied whenever there is only one
Lorentz invariant for a given partitioning of derivatives among the fields. At
higher numbers of derivatives, equation of motion redundancies can be removed,
but the increased number of Lorentz contractions spoils the subtraction of
integration by parts redundancies. While restricted, this technique is
sufficient to automatically generate the complete set of invariant operators of
the Standard Model effective field theory for dimensions 6 and 7 (for arbitrary
numbers of flavors). At dimension 8, the algorithm does not automatically
generate the complete operator set; however, it suffices for all but five
classes of operators. For these remaining classes, there is a well defined
procedure to manually determine the number of invariants. Using these methods,
we thereby derive the set of 535 dimension-8 $N_f = 1$ operators.
