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.

EFEKTIVITAS OPAC SEBAGAI SARANA TEMU KEMBALI INFORMASI DI UPT PERPUSTAKAAN UNIVERSITAS MUHAMMADIYAH MAGELANG (UNIMMA)

Evaluation of information retrieval tool is very important to find out how far the retrieval system has worked. This study evaluates the OPAC of Muhammadiyah University of Magelang (UNIMMA) by determining the effectiveness of the information retrieval system seen from the relevance of the data displayed with the data requested by the user. This is descriptive quantitative research that calculates the value of recall and precision, while the data collection is conducted through literature studies and documentation. Based on the data collected, it can be concluded that OPAC UNIMMA is in the effective category with a high level of precision, and in the value range of 0.68 - 1.00 with recall and precision values respectively 0.77 or 77% and 0.84 or 84%. However, from the observations on each class number, it is found that the class number has a higher recall value than the precision, ie: the class numbers of 200, 800, and 900.

NOTE ON THE p-DIVISIBILITY OF CLASS NUMBERS OF AN INFINITE FAMILY OF IMAGINARY QUADRATIC FIELDS

For any odd prime p, we construct an infinite family of imaginary quadratic fields whose class numbers are divisible by p. We give a corollary that settles Iizuka’s conjecture for the case n=1 and p>2.