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∪ S∪ 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.