On the biases and asymptotics of partitions with finite choices of parts

Biases in integer partitions have been studied recently. For three disjoint subsets $R,S,I$ of positive integers, let $p_{RSI}(n)$ be the number of partitions of $n$ with parts from $R\cup S\cup I$ and $p_{R>S,I}(n)$ be the number of such partitions with more parts from $R$ than that from $S$. In this paper, in the case that $R,S,I$ are finite we obtain a concrete formula of the asymptotic ratio of $p_{R>S,I}(n)$ to $p_{RSI}(n)$. We also propose a conjecture in the case that $R,S$ are certain infinite arithmetic progressions.