House monotonicity is the property of an apportionment method that increasing the house size while holding all populations
fixed cannot cause any state to lose a seat. For a population vector 
, let 
denote the number of seats assigned to state 
when the house size is 
. House monotonicity is equivalent to the inequalities
 |
for every state 
and house size 
.
An apportionment method satisfying these inequalities is called house monotone. The Alabama paradox occurs precisely when an apportionment method violates house monotonicity.
