## Hyunsoo Cho ; Byungchan Kim ; Hayan Nam ; Jaebum Sohn - A survey on t-core partitions

hrj:8928 - Hardy-Ramanujan Journal, January 9, 2022, Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021 - https://doi.org/10.46298/hrj.2022.8928
A survey on t-core partitions

Authors: Hyunsoo Cho ; Byungchan Kim ; Hayan Nam ; Jaebum Sohn

$t$-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.

Volume: Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021
Published on: January 9, 2022
Accepted on: January 9, 2022
Submitted on: January 6, 2022
Keywords: t-core,modular equation,modular forms,simultaneous core,crank,lattice path,2010 Mathematics Subject Classification. 11P81, 11P82; 11P83,[MATH]Mathematics [math]