  In this paper we study the Theta splitting function $\Theta(s+1)$, a function
defined on the positive integers. We study the distribution of this function
for sufficiently large values of the integers. As an application we show that
\begin{align}\sum \limits_{m=0}^{s}\prod \limits_{\substack{0\leq j \leq
m\\\sigma:[0,m]\rightarrow [0,m]\\\sigma(j)\neq \sigma(i)}}(s-\sigma(j))\sim
s^s\sqrt{s}e^{-s}\sum
\limits_{m=1}^{\infty}\frac{e^m}{m^{m+\frac{1}{2}}}.\nonumber \end{align} and
that \begin{align}\sum \limits_{j=0}^{s-1}e^{-\gamma j}\prod
\limits_{m=1}^{\infty}\bigg(1+\frac{s-j}{m}\bigg)e^{\frac{-(s-j)}{m}}\sim
\frac{e^{-\gamma s}}{\sqrt{2\pi}}\sum
\limits_{m=1}^{\infty}\frac{e^m}{m^{m+\frac{1}{2}}}.\nonumber \end{align}
