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

TL;DR

This paper investigates biases in the distribution of parts in k-indivisible partitions across congruence classes, revealing an ordering that aligns with natural order when k is sufficiently large, using Wright's circle method.

Contribution

It introduces the concept of the k-indivisible ordering of congruence classes and establishes conditions under which this ordering coincides with the natural order.

Findings

Asymptotic distribution of parts shows weak equidistribution among classes.

Biases create a specific ordering of congruence classes, called the k-indivisible ordering.

For large k, the ordering matches the natural order.

Abstract

For integers $k,t \geq 2$, and $1\leq r \leq t$ 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 $k \geq \frac{6(t^2-1)}{\pi^2}$ the $k$-indivisible ordering matches the natural ordering. We also explore the properties of these orderings when $k < \frac{6(t^2-1)}{\pi^2}$.