A survey on t-core partitions

t-core partitions have played important roles in the theory of partitions and related areas. In this survey, we briefly summarize interesting and important results on t-cores from classical results like how to obtain a generating function to recent results like simultaneous cores. Since there have been numerous studies on t-cores, it is infeasible to survey all the interesting results. Thus, we mainly focus on the roles of t-cores in number theoretic aspects of partition theory. This includes the modularity of t-core partition generating functions, the existence of t-core partitions, asymptotic formulas and arithmetic properties of t-core partitions, and combinatorial and number theoretic aspects of simultaneous core partitions. We also explain some applications of t-core partitions, which include relations between core partitions and self-conjugate core partitions, a t-core crank explaining Ramanujan’s partition congruences, and relations with class numbers.