Unexpected biases between congruence classes for in k-indivisible

For integers k, t > 2, and 1 < r < t let Dkx (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 Dkx (r, t; n) as n-+ oO 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 > 6(t2-1) pi 2 the k -indivisible ordering matches the natural ordering. We also explore the properties of these orderings when k < 6(t2-1) pi 2 .(c) 2023 Elsevier Inc. All rights reserved.