  We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture
confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose
gas with repulsive hard-core interactions) and of a non-interacting Fermi gas
(1D spin-aligned Fermi gas), both species interacting through hard-core
repulsive interactions. Using a generalized Bose-Fermi mapping, we determine
the exact particle density profiles, momentum distributions and behaviour of
the mixture under 1D expansion when opening the trap. In real space, bosons and
fermions do not display any phase separation: the respective density profiles
extend over the same region and they both present a number of peaks equal to
the total number of particles in the trap. In momentum space the bosonic
component has the typical narrow TG profile, while the fermionic component
shows a broad distribution with fermionic oscillations at small momenta. Due to
the large boson-fermion repulsive interactions, both the bosonic and the
fermionic momentum distributions decay as $C p^{-4}$ at large momenta, like in
the case of a pure bosonic TG gas. The coefficient $C$ is related to the
two-body density matrix and to the bosonic concentration in the mixture. When
opening the trap, both momentum distributions "fermionize" under expansion and
turn into that of a Fermi gas with a particle number equal to the total number
of particles in the mixture.
