Congruences for Bipartitions with Odd Parts Distinct

TL;DR

This paper investigates the arithmetic properties of bipartitions with odd parts distinct, deriving new identities and divisibility results, and providing combinatorial interpretations for these congruences.

Contribution

It introduces two Ramanujan-type identities for the bipartition function with odd parts distinct and establishes new divisibility properties along with combinatorial interpretations.

Findings

$pod_{-2}(2n+1)$ is even

$pod_{-2}(3n+2)$ is divisible by 3

Certain $pod_{-2}$ values are divisible by 3 or 5 for specific forms

Abstract

Hirschhorn and Sellers studied arithmetic properties of the number of partitions with odd parts distinct. In another direction, Hammond and Lewis investigated arithmetic properties of the number of bipartitions. In this paper, we consider the number of bipartitions with odd parts distinct. Let this number be denoted by $pod_{-2}(n)$. We obtain two Ramanujan type identities for $pod_{-2}(n)$, which imply that $pod_{-2}(2n+1)$ is even and $pod_{-2}(3n+2)$ is divisible by 3. Furthermore, we show that for any $\alpha\geq 1$ and $n\geq 0$, $ pod_{-2}(3^{2\alpha+1}n+\frac{23\times 3^{2\alpha}-7}{8})$ is a multiple of 3 and $pod_{-2}(5^{\alpha+1}n+\frac{11\times 5^\alpha+1}{4})$ is divisible by 5. We also find combinatorial interpretations for the two congruences modulo 2 and 3.