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 $\equiv \pm i(\mathrm{mod}k)$ may be overlined. We study the parity and distribution results for ${\overline{C}}_{k,i}(n),$ where $k>3$ and $1\le i\le \lfloor \frac{k}{2}\rfloor$. More particularly, we prove that for each integer $\ell \ge 2$ depending on $k$ and $i$, the interval $\left[\ell ,\frac{\ell (3\ell +1)}{2}\right]$ $($resp.\ $\left[2\ell -1,\frac{\ell (3\ell -1)}{2}\right])$ 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 $p\ge 5$ be a prime number.

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