  We discuss in this paper combinatorial aspects of boundary loop models, that
is models of self-avoiding loops on a strip where loops get different weights
depending on whether they touch the left, the right, both or no boundary. These
models are described algebraically by a generalization of the Temperley-Lieb
algebra, dubbed the two-boundary TL algebra. We give results for the dimensions
of TL representations and the corresponding degeneracies in the partition
functions. We interpret these results in terms of fusion and in the light of
the recently uncovered A_n large symmetry present in loop models, paving the
way for the analysis of the conformal field theory properties. Finally, we
propose conjectures for determinants of Gram matrices in all cases, including
the two-boundary one, which has recently been discussed by de Gier and Nichols.
