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For integers $k,t \geq 2$, and $1\leq r \leq t$ let $D_k^\times(r,t;n)$ bethe 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 congruentto $r$ 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 ofthis asymptotic does not depend on $r$, and so, in a weak asymptotic sense, theparts are equidistributed among congruence classes. However, inspection of thelower order terms indicates a bias towards different congruence classes modulo$t$. This induces an ordering on the congruence classes modulo $t$, which wecall the $k$-indivisible ordering. We prove that for $k \geq\frac{6(t^2-1)}{\pi^2}$ the $k$-indivisible ordering matches the naturalordering. We also explore the properties of these orderings when $k <\frac{6(t^2-1)}{\pi^2}$.

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