Some results concerning partitions with designated summands

Let PD(n) and PDO(n) count, respectively, the number of partitions of n with designated summands and the number of partitions of n with designated summands where all parts are odd, and let PDt(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$PD_t(n)$$\end{document} and PDOt(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$PDO_t(n)$$\end{document} count, respectively, the number of tags (that is, designated summands) in the partitions enumerated by PD(n) and PDO(n). We give elementary proofs of congruences for these partition functions.