A LAnKe (also known as a Filippov algebra or a Lie algebra of the n-th kind) is a vector space equipped with a skew-symmetric n-linear form that satisfies the generalized Jacobi identity. Friedmann, Hanlon, Stanley and Wachs have shown that the symmetric group acts on the multilinear part of the free LAnKe on 2n-1 generators as an irreducible representation. They announced that the multilinear component on 3n-2 generators decomposes as a direct sum of two irreducible symmetric group representations and a proof was given recently in a subsequent paper by Friedmann, Hanlon and Wachs. In the present paper we provide a proof of the later statement. The two proofs are substantially different.

A LAnKe (also known as a Lie algebra of the th kind, or a Filippov algebra) is a vector space equipped with a skew-symmetric -linear form that satisfies the generalized Jacobi identity. The symmetric group  acts on the multilinear part of the free LAnKe on  generators, where  is the number of brackets, by permutation of the generators. The corresponding representation was studied by Friedmann, Hanlon, Stanley and Wachs, who asked whether for , its irreducible decomposition contains no summand whose Young diagram has at most  columns. The answer is affirmative if $k \le 3$. In this paper, we show that the answer is affirmative for all $k$