  In this paper, we investigate how to quantify the quantum states of
$n$-particles from the point of $(k+1)$-partite entanglement $(1\leq k\leq
n-1)$, which plays an instrumental role in quantum nonlocality and quantum
metrology. We put forward two families of entanglement measures termed
$q$-$(k+1)$-PE concurrence $(q>1)$ and $\alpha$-$(k+1)$-PE concurrence
$(0\leq\alpha<1)$, respectively. As far as the pure state is concerned, they
are defined based on the minimum in entanglement. Meanwhile, rigorous proofs
showing that both types of quantifications fulfill all the requirements of an
entanglement measure are provided. In addition, we also propose two alternative
kinds of entanglement measures, named $q$-$(k+1)$-GPE concurrence $(q>1)$ and
$\alpha$-$(k+1)$-GPE concurrence $(0\leq\alpha<1)$, respectively, where the
quantifications of any pure state are given by taking the geometric mean of
entanglement under all partitions satisfying preconditions. Besides, the lower
bounds of these measures are presented by means of the entanglement of
permutationally invariant (PI) part of quantum states and the connections of
these measures are offered. Moreover, we compare these measures and explain the
similarities and differences among them. Furthermore, for computational
convenience, we consider enhanced versions of the above quantifications that
can be utilized to distinguish whether a multipartite state is genuinely strong
$k$-producible.
