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 nth Catalan number. This leads to a more general result on eigenspaces of a certain linear operator. A decomposition, into irreducibles, of the representation of S_3n-2 on the multilinear component the free LAnKe with 3n-2 generators is also presented. We also obtain a new presentation of Specht modules of shape λ, where λ has strictly decreasing column lengths, as a consequence of our eigenspace result.