  In this work we consider the permutational properties of multipartite
entanglement monotones. Based on the fact that genuine multipartite
entanglement is a property of the entire multi-qubit system, we argue that
ideal definitions for its characterizing quantities must be
permutation-invariant. Using this criterion, we examine the three 4-qubit
entanglement monotones introduced by Osterloh and Siewert [Phys. Rev. A. 72,
012337]. By expressing them in terms of quantities whose permutational
properties can be easily derived, we find that one of these monotones is not
permutation-invariant. We propose a permutation-invariant entanglement monotone
to replace it, and show that our new monotone properly measures the genuine
4-qubit entanglement in 4-qubit cluster-class states. Our results provide some
useful insights in understanding multipartite entanglement.
