Na Chen, Xiaorong Li, Olivia X.M. Yao: Infinite families of congruences modulo 9 for 9-regular partitions , 163-172

Abstract:

For any integer $t \geq 2$, let $b_t(n)$ denote the number of $t$-regular partitions of $n$. Recently, infinite families of congruences modulo 3 for $b_9(n)$ were discovered by Cui, Gu, Keith and Yao. In this paper, we establish infinite families of congruences modulo 9 for $b_9(n)$. We prove that for any prime $p$ with $p\equiv 1\ ({\rm mod}\ 6)$, there exists an integer $c(p)\in \{2,\ 3,\ 4\}$ such that for $n,\ \alpha\geq 0$, if $p\nmid (6n+5)$, then $b_9\left( 4p^{c(p)\alpha+c(p)-1}n+\frac{10p^{c(p)\alpha+c(p)-1}-1}{3 }\right... ...(p)-1}n+\frac{40p^{c(p)\alpha+c(p)-1}-1}{3 }\right)\equiv 0\ ({\rm mod}\ 9).$ Moreover, we prove some nonstandard congruences modulo 9 for $b_9(n)$. For example, we prove that for $\alpha\geq 0$, $b_9\left(\frac{10\times 7^{4\alpha}-1}{3}\right)\equiv - b_9\left(\frac{40\times 7^{4\alpha}-1}{3}\right)\equiv 3\times 2^{\alpha}\ ({\rm mod}\ 9).$

Key Words: Partition, regular partition, congruence.

2010 Mathematics Subject Classification: Primary 11P83; Secondary 05A17.