A note on the exact formulas for certain $2$-color partitions

Russelle Guadalupe

Published 2024-04-02Version 1

Let $p\leq 23$ be a prime and $a_p(n)$ count the number of partitions of $n$ using two colors where one of the colors only has parts divisible by $p$. Using a result of Sussman, we derive the exact formula for $a_p(n)$ and obtain an asymptotic formula for $\log a_p(n)$. Our results partially extend the work of Mauth, who proved the asymptotic formula for $\log a_2(n)$ conjectured by Banerjee et al.