2 problems

Let $f_m:=\prod_{j=1}^{\infty}(1-q^{jm})$ for every positive integer $m$, and let $\text{PDO}_t(n)$ denote the total number of tagged parts over all partitions of $n$ with designat…

A partition with designated summands is obtained from an ordinary partition by tagging exactly one occurrence of each part size. Let $\text{PDO}_t(n)$ denote the total number of ta…