Title: On a generalization of Lie($k$): a CataLAnKe theorem,
Abstract: We define a generalization of the free Lie algebra based on an $n$-ary
commutator and call it the free LAnKe. We show that the action of the symmetric
group $S_{2n-1}$ on the multilinear component with $2n-1$ generators is given
by the representation $S^{2^{n-1}1}$, whose dimension is the $n$th Catalan
number. An application involving Specht modules of staircase shape is
presented. We also introduce a conjecture that extends the relation between the
Whitehouse representation and Lie($k$).