  In the classic apportionment problem the goal is to decide how many seats of
a parliament should be allocated to each party as a result of an election. The
divisor methods provide a way of solving this problem by defining a notion of
proportionality guided by some rounding rule. Motivated by recent challenges in
the context of electoral apportionment, we consider the question of how to
allocate the seats of a parliament under parity constraints between candidate
types (e.g. equal number of men and women elected) while at the same time
satisfying party proportionality.
  We consider two different approaches for this problem. The first mechanism,
that follows a greedy approach, corresponds to a recent mechanism used in the
Chilean Constitutional Convention 2021 election. We analyze this mechanism from
a theoretical point of view. The second mechanism follows the idea of
biproportionality introduced by Balinski and Demange [Math. Program. 1989,
Math. Oper. Res. 1989]. In contrast with the classic biproportional method by
Balinski and Demange, this mechanism is ruled by two levels of proportionality:
Proportionality is satisfied at the level of parties by means of a divisor
method, and then biproportionality is used to decide the number of candidates
allocated to each type and party. We provide a theoretical analysis of this
mechanism, making progress on the theoretical understanding of methods with two
levels of proportionality. A typical benchmark used in the context of
two-dimensional apportionment is the fair share (a.k.a matrix scaling), which
corresponds to an ideal fractional biproportional solution. We provide lower
bounds on the distance between these two types of solutions, and we explore
their consequences in the context of two-dimensional apportionment.
