Distribution of Andrews’ singular overpartitions ${\overline{C}}_{p,1}(n)$

Abstract

Andrews introduced the partition function ${\overline{C}}_{k, i}(n)$, called singular overpartition, which counts the number of overpartitions of n in which no part is divisible by k and only parts $\equiv \pm i\pmod {k}$ may be overlined. We study the parity and distribution results for ${\overline{C}}_{k,i}(n),$ where $k>3$ and $1\le i \le \left\lfloor \frac{k}{2}\right\rfloor$. More particularly, we prove that for each integer $\ell \ge 2$ depending on k and i, the interval $\left[ \ell , \frac{\ell (3\ell +1)}{2}\right]$ $\Big ($resp. $\left[ 2\ell -1, \frac{\ell (3\ell -1)}{2}\right] \Big )$ contains an integer n such that ${\overline{C}}_{k,i}(n)$ is even (resp. odd). Finally we study the distribution for ${\overline{C}}_{p,1}(n)$ where $p\ge 5$ is a prime number.