# Some congruences for $(s,t)$-regular bipartitions modulo $t$

## TL;DR

This paper investigates congruences for the function counting $(s,t)$-regular bipartitions, extending known results by proving new infinite families of congruences modulo various primes for specific $(s,t)$ pairs.

## Contribution

It establishes new infinite families of congruences for $B_{s,t}(n)$ for various $(s,t)$, expanding the understanding of bipartition congruences beyond previous results.

## Key findings

- Proves congruences modulo 5 for $B_{2,15}(n)$.
- Establishes congruences modulo 11 for $B_{7,11}(n)$ and $B_{27,11}(n)$.
- Demonstrates congruences modulo 17 for $B_{243,17}(n)$.

## Abstract

In this work, we study the function $B_{s,t}(n)$, which counts the number of $(s,t)$-regular bipartitions of $n$. Recently, many authors proved infinite families of congruences modulo $11$ for $B_{3,11}(n)$, modulo $3$ for $B_{3,s}(n)$ and modulo $5$ for $B_{5,s}(n)$. Very recently, Kathiravan proved several infinite families of congruences modulo $11$, $13$ and $17$ for $B_{5,11}(n)$, $B_{5,13}(n)$ and $B_{81,17}(n)$. In this paper, we will prove infinite families of congruences modulo $5$ for $B_{2,15}(n)$, modulo $11$ for $B_{7,11}(n)$, modulo $11$ for $B_{27,11}(n)$ and modulo $17$ for $B_{243,17}(n)$.