  We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of
finite length N. For the case that q is a root of unity, we propose a formula
for the number of admissible solutions of the Bethe ansatz equations in terms
of dimensions of irreducible representations of the Temperley-Lieb algebra; and
a formula for the degeneracies of the transfer matrix eigenvalues in terms of
dimensions of tilting sl(2)_q-modules. These formulas include corrections that
appear if two or more tilting modules are spectrum-degenerate. For the XX case
(q=exp(i pi/2)), we give explicit formulas for the number of admissible
solutions and degeneracies. We also consider the cases of generic q and the
isotropic (q->1) limit. Numerical solutions of the Bethe equations up to N=8
are presented. Our results are consistent with the Bethe ansatz solution being
complete.
