  We investigate the spin-polarized chain of ultracold fermionic atoms with
spin-3/2 described by the fermionic Hubbard model with SU(4) symmetric
attractive interaction. The competition of bound pairs, trions, quartets and
unbound atoms is studied analytically and by density matrix renormalization
group simulations. We find several distinct states where bound particles
coexist with the ferromagnetic state of unpaired fermions. In particular, an
exotic inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-type superfluid of
quartets in a magnetic background of uncorrelated atoms is found for weaker
interactions. We show that the system can be driven from this quartet-FFLO
state to a molecular state of localized quartets which is also reflected in the
static structure factor. For strong enough coupling, spatial segregation
between molecular crystals and ferromagnetic liquids emerges due to the large
effective mass of the composite particles.
