Arithmetic Identities and Congruences for Partition Triples with 3-cores

Abstract: Let ${{B}{3}}(n)$ denote the number of partition triples of $n$ where each partition is 3-core. With the help of generating function manipulations, we find several infinite families of arithmetic identities and congruences for ${{B}{3}}(n)$. Moreover, let $\omega (n)$ denote the number of representations of a nonnegative integer $n$ in the form $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+3y_{1}^{2}+3y_{2}^{2}+3y_{3}^{2}$ with ${{x}{1}},{{x}{2}},{{x}{3}},{{y}{1}},{{y}{2}},{{y}{3}}\in \mathbb{Z}.$ We find three arithmetic relations between ${{B}{3}}(n)$ and $\omega (n)$, such as $\omega (6n+5)=4{{B}{3}}(6n+4).$