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 pRSI(n) be the number of partitions of n with parts from R∪ S∪ I and pR>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 pR>S,I(n) to pRSI(n). We also propose a conjecture in the case that $R,S$ are certain infinite arithmetic progressions.