  We focus on a uniform partition problem in a population protocol model. The
uniform partition problem aims to divide a population into k groups of the same
size, where k is a given positive integer. In the case of k=2 (called uniform
bipartition), a previous work clarified space complexity under various
assumptions: 1) an initialized base station (BS) or no BS, 2) weak or global
fairness, 3) designated or arbitrary initial states of agents, and 4) symmetric
or asymmetric protocols, except for the setting that agents execute a protocol
from arbitrary initial states under weak fairness in the model with an
initialized base station. In this paper, we clarify the space complexity for
this remaining setting. In this setting, we prove that P states are necessary
and sufficient to realize asymmetric protocols, and that P+1 states are
necessary and sufficient to realize symmetric protocols, where P is the known
upper bound of the number of agents. From these results and the previous work,
we have clarified the solvability of the uniform bipartition for each
combination of assumptions. Additionally, we newly consider an assumption on a
model of a non-initialized BS and clarify solvability and space complexity in
the assumption. Moreover, the results in this paper can be applied to the case
that k is an arbitrary integer (called uniform k-partition).
