  We investigate the ground state of the one-dimensional fermionic system
enclosed in a hard-wall trap with attractive contact p-wave interactions. Based
on the Bethe ansatz method, the explicit wave function is derived by
numerically solving the Bethe ansatz equations for the full physical regimes
($-\infty \leq c_F\leq 0$). With the exact wave function some quantities which
are important in many-body physics are obtained, including the one-body density
matrix and the momentum distribution of the ground state for finite system. It
is shown that the shell structure of the density profiles disappears with the
increase of the interaction and in the fermionic Tonks-Girardeau (FTG) limit
the density distribution shows the same behavior as that of an ideal Bose gas.
However the one-body density matrix and the momentum distribution exhibit
completely different structures compared with their bosonic counterparts.
