  We study a minimal model of active transport in crowded single-file
environments which generalises the emblematic model of single file diffusion to
the case when the tracer particle (TP) performs either an autonomous directed
motion or is biased by an external force, while all other particles of the
environment (bath) perform unbiased diffusions. We derive explicit expressions,
valid in the limit of high density of bath particles, of the full distribution
$P_n(X)$ of the TP position and of all its cumulants, for arbitrary values of
the bias $f$ and for any time $n$. Our analysis reveals striking features, such
as the anomalous scaling $\propto\sqrt{n}$ of all cumulants, the equality of
cumulants of the same parity characteristic of a Skellam distribution and a
convergence to a Gaussian distribution in spite of asymmetric density profiles
of bath particles. Altogether, our results provide the full statistics of the
TP position, and set the basis for a refined analysis of real trajectories of
active particles in crowded single-file environments.
