  We use Constraint Satisfaction Methods to construct and enumerate finite
$L$-algebras up to isomorphism. These objects were recently introduced by Rump
and appear in Garside theory, algebraic logic, and the study of the
combinatorial Yang-Baxter equation. There are 377322225 isomorphism classes of
$L$-algebras of size eight. The database constructed suggest the existence of
bijections between certain classes of $L$-algebras and well-known combinatorial
objects. On the one hand, we prove that Bell numbers enumerate isomorphism
classes of finite linear $L$-algebras. On the other hand, we also prove that
finite regular $L$-algebras are in bijective correspondence with
infinite-dimensional Young diagrams.
