Distribution of Andrews' Singular Overpartitions ${\bar{C}}_{p,1}(n)$

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