On Multirank For Certain Colored Partitions

The paper first introduces the integer partitions in combinatorics of the branch of mathematics,which we focuses on two topics: partition identities and partition congruences.In the preliminary knowledge,we introduce two kinds of tools that are commonly used in the study of integer partitions: the generating function that is suitable for algebraic manipulation,and the Ferrers' graph that has geometric intuition.We give examples to demonstrate how these two methods are used repectively.Then we review the knowledge about the minimal polynomial and the primitive root of unity.Then,in the third part of the paper,we first review two classic proofs of the Ramanujan congruences,especially the one where Garvan utilized the crank statistics to give a proof of the refined results.Which inspires us to research a kind of special colored partition and its congruences properties and we use a similar approach defining multirank and give the combinatorial interpretation of congruences.When t = 3 this conclusion was first given by Reti [36].Finally,we give a summary of the entire paper,and mention the problems that merit further investigation.