An simplified example of my model structure would be
class Corporation(models.Model):
...
class Division(models.Model):
corporation = models.ForeignKey(
You should be able to do this with a Case() expression to query the count of departments that have the type you are looking for:
from django.db.models import Case, IntegerField, Sum, When, Value
Corporation.objects.annotate(
type_10_count=Sum(
Case(
When(division__department__type=10, then=Value(1)),
default=Value(0),
output_field=IntegerField()
)
)
)