Commun. Korean Math. Soc. 2024; 39(2): 345-352
Online first article April 24, 2024 Printed April 30, 2024
https://doi.org/10.4134/CKMS.c230214
Copyright © The Korean Mathematical Society.
Byungchan Kim
Seoul National University of Science and Technology
We introduce a new class of bi-partition function $c_k(n)$, which counts the number of bi-color partitions of $n$ in which the second color only appears at the parts that are multiples of $k$. We consider two partitions to be the same if they can be obtained by switching the color of parts that are congruent to zero modulo $k$. We show that the generating function for $c_k(n)$ involves the partial theta function and obtain the following congruences: \begin{align*} c_2 (27n+26) &\equiv 0 \pmod{3} \\ \intertext{and} c_3 (4n + 2 ) &\equiv 0 \pmod{2}. \end{align*}
Keywords: Color change, bi-partition, partial theta function, congruence
MSC numbers: Primary 11P81, 11P83
Supported by: This study was supported by the Research Program funded by the SeoulTech(Seoul National University of Science and Technology)..
2014; 29(1): 1-8
2013; 28(3): 433-447
2009; 24(2): 181-185
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd