  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.
