fields in the EFT are bosonic. Denoting the (possibly infinite) set of generators of the EFT basis by $X_{\text{gen}}$, the denominator of the Hilbert series, $H = N/D$, takes the form $$D(t,\{u_i\}) = \prod_a (1-h_a),$$ where $h_a$ is the weight of an operator $\scO_a \in X_{\text{gen}}$. In the case where any operator in the basis can be *uniquely* written as a polynomial of the generators we say the basis is *freely generated*, *i.e.* there are no non-trivial relations among the generators. In this case the numerator is unity: binomial expansion of the denominator clearly shows that the weights of all operators in the basis are obtained and counted once. A numerator of $H$ differing from unity accounts for relations among the generators.

The grading where each field has a unique weight is, of course, just a choice; while this grading retains the most information about the basis, different weighting choices may be more expedient for other purposes. For example, we could grade by dimension of operators: let $d_i = [\ph_i]$ be the dimension of $\ph_i$ and send $u_i \to q^{d_i}$ and $t \to q$ so that the coefficient $c_k$ in expansion of the Hilbert series, $H(t,\{u_i\}) \to H(q) = \sum 