Unexpected Biases between Congruence Classes for Parts in k-indivisible Partitions

Abstract

For integers $k,t\ge 2$, and $1\le r\le t$ let $D_k^{\times }(r,t;n)$ be the number of parts among all $k$-indivisible partitions of $n$ (i.e., partitions where all parts are not divisible by $k$) of $n$ that are congruent to $r$ modulo $t$. Using Wright's circle method, we derive an asymptotic for $D_k^{\times }(r,t;n)$ as $n\to \infty$ when $k,t$ are coprime. The main term of this asymptotic does not depend on $r$, and so, in a weak asymptotic sense, the parts are equidistributed among congruence classes. However, inspection of the lower order terms indicates a bias towards different congruence classes modulo $t$. This induces an ordering on the congruence classes modulo $t$, which we call the $k$-indivisible ordering. We prove that for $k\ge \frac{6(t^2-1)}{{\pi }^2}$ the $k$-indivisible ordering matches the natural ordering. We also explore the properties of these orderings when $k<\frac{6(t^2-1)}{{\pi }^2}$.

26 pages, 3 figures