  We derive all-order expressions for perturbations of the Einstein-Hilbert
action and the Einstein equation with the general $n$-th order terms. To this
end, we employ Cheung and Remmen's perturbation conventions both in tensor
density and the usual metric tensor formalisms, including the Einstein-dilaton
theory. Remarkably, we find minimal building blocks that generate the entire
perturbations for each of our formulations. We show that the number of terms of
perturbations grows linearly as the order of perturbations increases. We regard
our results as the reference and discuss how to derive perturbations in other
conventions from the reference. As a consistency check, we compute graviton
scattering amplitudes using the perturbiner method based on the perturbative
Einstein equation. Finally we discuss how to generalise the results to curved
backgrounds and incorporate additional matter.
