Biases among Congruence Classes for Parts in k-regular Partitions

Abstract

For integers $k,t\ge 2$ and $1\le r\le t$ 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 \[ D_{k}(r,t;n) = \frac{3^{\frac{1}{4}}e^{π\sqrt{\frac{2Kn}{3}}}}{πt 2^{\frac{3}{4}}K^{\frac{1}{4}}n^{\frac{1}{4}}\sqrt{k}}\left(\log k + \left(\frac{3\sqrt{K}\log k}{8\sqrt{6}π} - \frac{tπ(k-1)K^{\frac{1}{2}}}{2\sqrt{6}}\left(\frac{r}{t}- \frac{1}{2}\right)\right)n^{-\frac{1}{2}} + O(n^{-1})\right), \] 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 $1\le r<s\le t$ we have $D_k(r,t;n)\ge D_k(s,t;n)$ for sufficiently large $n$. We make this inequality explicit, showing that for $3\le k\le 10$ and $2\le t\le 10$ the inequality $D_k(r,t;n)\ge D_k(s,t;n)$ holds for all $n\ge 1$ and the strict inequality $D_k(r,t;n)>D_k(s,t;n)$ holds for all $n\ge 17$.

25 pages, 3 figures