  The paper addresses the single-file diffusion in the presence of an absorbing
boundary. The emphasis is on an interplay between the hard-core interparticle
interaction and the absorption process. The resulting dynamics exhibits several
qualitatively new features. First, starting with the exact probability density
function for a given particle (a tracer), we study the long-time asymptotics of
its moments. Both the mean position and the mean square displacement are
controlled by dynamical exponents which depend on the initial order of the
particle in the file. Secondly, conditioning on non-absorption, we study the
distribution of long-living particles. In the conditioned framework, the
dynamical exponents are the same for all particles, however, a given particle
possesses an effective diffusion coefficient which depends on its initial
order. After performing the thermodynamic limit, the conditioned dynamics of
the tracer is subdiffusive, the generalized diffusion coefficient $D_{1/2}$
being different from that reported for the system without absorbing boundary.
