  We consider bosons at Landau level filling $\nu=1$ on a thin torus. In
analogy with previous work on fermions at filling $\nu =1/2$, we map the
low-energy sector onto a spin-1/2 chain. While the fermionic system may realize
the gapless XY-phase, we show that typically this does not happen for the
bosonic system. Instead, both delta function and Coulomb interaction lead to
gapped phases in the bosonic system, and in particular we identify a phase
corresponding to the non-abelian Moore-Read state. In the spin language, the
hamiltonian is dominated by a ferromagnetic next-nearest neighbor interaction,
which leads to a description consistent with the non-trivial degeneracies of
the ground and excited states of this phase of matter. In addition we comment
on the similarities and differences of the two systems mentioned above and
fermions at $\nu=5/2$.
