Notes on the $$\textit{spt}$$ spt function of George E. Andrews

Andrews defined $$\textit{spt}(n)$$ to be the total number of appearances of the smallest parts in all of the partitions of $$n$$ . In this paper, we study the statistical distribution of $$\textit{spt}(\pi )$$ , the number of smallest parts in the partition $$\pi $$ as $$\pi $$ ranges over all partitions of $$n$$ . We also give a combinatorial proof of a conjecture of Hirschhorn, namely that $$\begin{aligned} p(0)+\ \cdots \ +p(n-1)1$$ , where $$p(n)$$ is the number of partitions of $$n$$ .