On the number of equivalence classes of bi-partitions arising from the color change

We introduce a new class of bi-partition function $c_k(n)$, which counts the number of bi-color partitions of $n$ in which the second color only appears at the parts that are multiples of $k$. We consider two partitions to be the same if they can be obtained by switching the color of parts that are congruent to zero modulo $k$. We show that the generating function for $c_k(n)$ involves the partial theta function and obtain the following congruences: \begin{align*} c_2 (27n+26) &\equiv 0 \pmod{3} \\ \intertext{and} c_3 (4n + 2 ) &\equiv 0 \pmod{2}. \end{align*} KCI Citation Count: 0