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

Abstract

Biases in integer partitions have been studied recently. For three disjoint subsets $R,S,I$ of positive integers, let $p_{RSI}\left(n\right)$ be the number of partitions of $n$ with parts from $R\cup S\cup I$ and $p_{R>S,I}\left(n\right)$ be the number of such partitions with a greater number of parts in $R$ than that in $S$. In this paper, in the case that $R,S,I$ are finite, we obtain an explicit formula of the asymptotic ratio of $p_{R>S,I}\left(n\right)$ to $p_{RSI}\left(n\right)$. The key technique for computing this ratio is to estimate a partition number at the volume of a certain polytope. A conjecture is proposed in the case that $R,S$ are certain infinite arithmetic progressions.