Papers › Biases among Congruence Classes for Parts in k-regular Partitions
Biases among Congruence Classes for Parts in k-regular Partitions
Faye Jackson, Misheel Otgonbayar
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For integers k,t ≥2 and 1≤r ≤t let Dₖ(r,t;n) be the number of parts among all k-regular partitions (i.e., partitions of n where all parts have multiplicity less than k) of n that are congruent to r modulo t. Using the circle method, we obtain the asymptotic Dₖ(r,t;n) = (3^(1/4)e^(π√(2Kn/3)))/(πt 2^(3/4)K^(1/4)n^(1/4)√(k))(logk + ((3√(K)logk)/(8√(6)π) - (tπ(k-1)K^(1/2))/(2√(6))(r/t- 1/2))n^(-1/2) + O(n⁻¹)), where K = 1 - 1/k. The main term of this asymptotic does not depend on r, and so if Pₖ(n) is the total number of parts among all k-regular partitions of n, we have that (Dₖ(r,t;n))/(Pₖ(n)) →1/t as n →∞. Thus, in a weak asymptotic sense, the parts are equidistributed among congruence classes. However, inspection of the lower order terms indicates a bias towards the lower congruence classes; that is, for 1≤r < s ≤t we have Dₖ(r,t;n) ≥Dₖ(s,t;n) for sufficiently large n. We make this inequality explicit, showing that for 3 ≤k ≤10 and 2 ≤t ≤10 the inequality Dₖ(r,t;n) ≥Dₖ(s,t;n) holds for all n ≥1 and the strict inequality Dₖ(r,t;n) > Dₖ(s,t;n) holds for all n ≥17.
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