New Congruences for an Analogue of Lin's Partition Triples

Abstract

Let $B(n)$ represent the number of partition triples $\pi=(\pi_1,\pi_2,\pi_3)$ of $n$, where $\pi_1$ and $\pi_2$ consists of distinct odd parts and every part of $\pi_3$ is divisible by $4$. This partition function was introduced by $\mathrm{R.\ Guadalupe}$ as a counterpart to Lin's restricted partition function $b(n)$. Among the key findings in the original study were several Ramanujan-type congruences for $B(n)$ modulo $2$, $3$, $5$, $7$, and $9$, revealing rich arithmetic properties of the function. Motivated by these results, we further investigate the arithmetic behavior of $B(n)$ and establish new congruence relations. The present study extends the investigation of the arithmetic properties of $B(n)$, with particular emphasis on its behavior $modulo \;4$. By employing generating function techniques and $q$-series identities, we derive several new Ramanujan-type congruences and infinite families of congruence relations for $B(n)$ modulo $4$.