  We investigate the transition of a quasi-one-dimensional few-boson system
from a weakly correlated to a fragmented and finally a fermionized ground
state. Our numerically exact analysis, based on a multi-configurational method,
explores the interplay between different shapes of external and inter-particle
forces. Specifically, we demonstrate that the addition of a central barrier to
an otherwise harmonic trap may supports the system's fragmentation, with a
symmetry-induced distinction between even and odd atom numbers. Moreover, the
impact of inhomogeneous interactions is studied, where the effective coupling
strength is spatially modulated. It is laid out how the ground state can be
displaced in a controlled way depending on the trap and the degree of
modulation. We present the one- and two-body densities and, beyond that,
highlight the role of correlations on the basis of the natural occupations.
