operators related by the EOM lead to the same physical effects [@Politzer:1980me; @Georgi:1991ch]. The EOM follow from the requirement =0 . The above equation is often evaluated at lowest order in an expansion parameter of the EFT; the operator relations implied are then worked out at each successive order. For the purpose of simply counting operators, considering equations of motion derived from kinetic terms is sufficient. We therefore have the following equivalence, $$\scO_a \sim \scO_b \quad \text{if} \quad \scO_a = \scO_b + \scO_c \frac{\d S_{\text{kin}}}{\d \ph_j},
    \label{eqn:equiv_EOM}$$ for some $\ph_j \in \{\ph_i\}$.

The physical system to which we restrict ourselves in this paper is a one-dimensional quantum field theory of $N$ real scalar fields, $\ph_1, \dots, \ph_N$. We refer to the $N$ fields as flavors and denote the generalized Hilbert series as $H_N(t,\{u_i\})$. Even though this is the simplest theory one could imagine for our purpose, we find surprising and non-trivial results when both IBP and EOM are accounted for.

The outline of this paper is as follows. In section \[sec:framework\] we develop a framework for computing the coefficients $c_{k\, r_1 \dots r_N}$ of the Hilbert series and constructing sets of independent operators, while taking into account IBP and EOM. 