In 2013, Bennett proposed a mereological theory in which the parthood relation is defined on the basis of two primitive relations: a is a part of b iff a fills a slot owned by b. However, this theory has issues counting how many parts an entity has. We explore the various counting problems and propose a new theory to solve them. Keeping the core idea of Bennett’s slots, this theory introduces mereological relations between slots. This theory enables us to solve (...) all known counting problems and to go beyond the limits of Bennett’s theory by theorising expected features of mereological theories: supplementation principles and mereological sum and fusion. The theory is illustrated on ontological issues on the nature of structural universals and informational entities. (shrink)