On a generalization of Lie(k): A CataLAnKe theorem

We initiate a study of the representation symmetric group on multilinear component an n-ary generalization free Lie algebra, which we call LAnKe. Our central result is that S2n−1 LAnKe with 2n−1 generators given by irreducible whose dimension nth Catalan number. This leads to more general eigenspaces certain linear operator, has additional consequences. also obtain new presentation Specht modules staircase shape as consequence our result.