  We analyze the non-abelian Read-Rezayi quantum Hall states on the torus,
where it is natural to employ a mapping of the many-body problem onto a
one-dimensional lattice model. On the thin torus--the Tao-Thouless (TT)
limit--the interacting many-body problem is exactly solvable. The Read-Rezayi
states at filling $\nu=\frac k {kM+2}$ are known to be exact ground states of a
local repulsive $k+1$-body interaction, and in the TT limit this is manifested
in that all states in the ground state manifold have exactly $k$ particles on
any $kM+2$ consecutive sites. For $M\neq 0$ the two-body correlations of these
states also imply that there is no more than one particle on $M$ adjacent
sites. The fractionally charged quasiparticles and quasiholes appear as domain
walls between the ground states, and we show that the number of distinct domain
wall patterns gives rise to the nontrivial degeneracies, required by the
non-abelian statistics of these states. In the second part of the paper we
consider the quasihole degeneracies from a conformal field theory (CFT)
perspective, and show that the counting of the domain wall patterns maps one to
one on the CFT counting via the fusion rules. Moreover we extend the CFT
analysis to topologies of higher genus.
