A survey on t-core partitions

Hyunsoo Cho (1) , Byungchan Kim , Hayan Nam , Jaebum Sohn

1 EWHA - EWHA Womans University

$t$-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.

Mots clés: t-core; modular equation; modular forms; simultaneous core; crank; lattice path

Domaines: Mathématiques [math]

Hyunsoo Cho, Byungchan Kim, Hayan Nam, Jaebum Sohn. A survey on t-core partitions. Hardy-Ramanujan Journal, 2022, Special Commemorative volume in honour of Srinivasa Ramanujan - 2021, Volume 44 - Special Commemorative volume in honour of Srinivasa Ramanujan - 2021, pp.81 -- 101. ⟨10.46298/hrj.2022.8928⟩. ⟨hal-03498194⟩