Biases among Congruence Classes for Parts in k-regular Partitions

Abstract

For integers $k,t \geq 2$ and $1\leq r \leq t$ let $D_k(r,t;n)$ 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 \[ D_{k}(r,t;n) = \frac{3^{\frac{1}{4}}e^{\pi\sqrt{\frac{2Kn}{3}}}}{\pi t 2^{\frac{3}{4}}K^{\frac{1}{4}}n^{\frac{1}{4}}\sqrt{k}}\left(\log k + \left(\frac{3\sqrt{K}\log k}{8\sqrt{6}\pi} - \frac{t\pi(k-1)K^{\frac{1}{2}}}{2\sqrt{6}}\left(\frac{r}{t}- \frac{1}{2}\right)\right)n^{-\frac{1}{2}} + O(n^{-1})\right), \] where $K = 1 - \frac{1}{k}$. The main term of this asymptotic does not depend on $r$, and so if $P_k(n)$ is the total number of parts among all $k$-regular partitions of $n$, we have that $\frac{D_k(r,t;n)}{P_k(n)} \to \frac{1}{t}$ as $n \to \infty$. 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 $1\leq r < s \leq t$ we have $D_k(r,t;n) \geq D_k(s,t;n)$ for sufficiently large $n$. We make this inequality explicit, showing that for $3 \leq k \leq 10$ and $2 \leq t \leq 10$ the inequality $D_k(r,t;n) \geq D_k(s,t;n)$ holds for all $n \geq 1$ and the strict inequality $D_k(r,t;n) > D_k(s,t;n)$ holds for all $n \geq 17$.