  The main focus of this thesis is the theoretical study of strongly
interacting quantum mixtures confined in one dimension and subjected to a
harmonic external potential. Such strongly correlated systems can be realized
and tested in ultracold atoms experiments. Their non-trivial permutational
symmetry properties are investigated, as well as their interplay with
correlations. Exploiting an exact solution at strong interactions, we extract
general correlation properties encoded in the one-body density matrix and in
the associated momentum distributions, in fermionic and Bose-Fermi mixtures. In
particular, we obtain substantial results about the short-range behavior, and
therefore the high-momentum tails, which display typical $k^{-4}$ laws. The
weights of these tails, denoted as Tan's contacts, are related to numerous
thermodynamic properties of the systems such as the two-body correlations, the
derivative of the energy with respect to the one-dimensional scattering length,
or the static structure factor. We show that these universal Tan's contacts
also allow to characterize the spatial symmetry of the systems, and therefore
is a deep connection between correlations and symmetries. Besides, the exchange
symmetry is extracted using a group theory method, namely the class-sum method,
which comes originally from nuclear physics. Moreover, we show that these
systems follow a generalized version of the famous Lieb-Mattis theorem. Wishing
to make our results as experimentally relevant as possible, we derive scaling
laws for Tan's contact as a function of the interaction, temperature and
transverse confinement. These laws display interesting effects related to
strong correlations and dimensionality.
