  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 component of the
free LAnKe on $2n-1$ generators as an irreducible representation. They have
claimed that the multilinear component on $3n-2$ generators decomposes as a
direct sum of two irreducible symmetric group representations. This paper
provides a proof of the later statement.
