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

2022/07/13 by Jackson, Faye, Otgonbayar, Misheel
#11P81 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary: 11P82. Secondary: 05A17

paper · doi:10.48550/arxiv.2207.06365

Abstract

For integers k,t ≥ 2, and 1≤ r ≤ t let D_k^×(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^×(r,t;n) as n → ∞ 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))/(π²) the k-indivisible ordering matches the natural ordering. We also explore the properties of these orderings when k < (6(t²-1))/(π²).

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