Covariance of scattering amplitudes from counting carefully

A bstract Invariance of on-shell scattering amplitudes under field redefinitions is a well known property in field theory that corresponds to covariance of on-shell amputated connected functions. In recent years there have been great efforts to define a formalism in which the covariance is manifest at all stages of calculation, mainly resorting to geometrical interpretations. In this work covariance is analysed using combinatorial methods relying only on the properties of the tree level effective action, without referring to specific formulations of the Lagrangian. We provide an explicit proof of covariance of on-shell connected functions and of the existence of covariant Feynman rules and we derive an explicitly covariant closed formula for tree level on-shell connected functions with any number of external legs.