  We introduce time-dependent, generalized factorial cumulants $C_s^m(t)$ of
the full counting statistics of electron transfer as a tool to detect
interactions in nanostructures. The violation of the sign criterion $(-1)^{m-1}
C^m_s(t)\ge0$ for \emph{any} time $t$, order $m$, and parameter $s$ proves the
presence of interactions. For given system parameters, there is a minimal time
span $t_\text{min}$ and a minimal order $m$ to observe the violation of the
sign criterion. We demonstrate that generalized factorial cumulants are more
sensitive to interactions than ordinary ones and can detect interactions even
in regimes where ordinary factorial cumulants fail. We illustrate our findings
with the example of a quantum dot tunnel coupled to electronic reservoirs
either in or out of equilibrium.
