### Abstract:

It is a classical result that the multilinear component of the free Lie algebra is isomorphic (as a representation of
the symmetric group) to the top (co)homology of the proper part of the poset of partitions \(\Pi_n\)
tensored with the sign representation. We generalize this result in order to study the multilinear
component of the free Lie algebra with multiple compatible Lie brackets. We introduce a new poset of weighted
partitions \(\Pi_n^k\) that allows us to generalize the result. The new poset is a generalization
of \(\Pi_n\) and of the poset of weighted partitions \(\Pi_n^w\) introduced by Dotsenko and Khoroshkin
and studied by the author and Wachs for the case of two compatible brackets. We prove that the
poset \(\Pi_n^k\) with a top element added is EL-shellable and hence Cohen-Macaulay.
This and other properties of \(\Pi_n^k\) enable us to answer
questions posed by Liu on free multibracketed Lie algebras. In
particular, we obtain various dimension formulas and multicolored generalizations of
the classical Lyndon and comb bases for the multilinear component of the free Lie
algebra.
We also obtain a plethystic formula for the
Frobenius characteristic of the representation of the symmetric group on the multilinear component
of the free multibracketed Lie algebra.