  We initiate a study of the representation of the symmetric group on the
multilinear component of an $n$-ary generalization of the free Lie algebra,
which we call a free LAnKe. Our central result is that the representation of
the symmetric group $S_{2n-1}$ on the multilinear component of the free LAnKe
with $2n-1$ generators is given by an irreducible representation whose
dimension is the $n$th Catalan number. This leads to a more general result on
eigenspaces of a certain linear operator, which has additional consequences. We
also obtain a new presentation of Specht modules of staircase shape as a
consequence of our central result.
