Some results concerning partitions with designated summands (Q829694)

The notion of partitions with designated summands was introduced by \textit{G. E.~Andrews} et al. [Acta Arith. 105, No. 1, 51--66 (2002; Zbl 1004.05006)]. Let \(\mathrm{PD}(n)\) and \(\mathrm{PDO}(n)\) denote 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. They derived some Ramanujan-type identities and congruences for \(\mathrm{PD}(n)\) and \(\mathrm{PDO}(n)\). For example, \begin{align*} \sum_{n=0}^\infty\mathrm{PD}(2n+1)q^n &=\frac{(q^2;q^2)_\infty^5(q^{12};q^{12})_\infty^2} {(q;q)_\infty^5(q^3;q^3)_\infty(q^4;q^4)_\infty^2(q^6;q^6)_\infty},\tag{1}\\ \sum_{n=0}^\infty\mathrm{PDO}(3n+2)q^n &=\frac{2(q^2;q^2)_\infty^3(q^6;q^6)_\infty (q^{12};q^{12})_\infty}{(q;q)_\infty^4(q^4;q^4)_\infty}.\tag{2} \end{align*} In 2013, \textit{W.Y.C.~Chen} et al. [J. Number Theory 133, No. 9, 2929--2938 (2013; Zbl 1295.05044)] further studied combinatorial and congruence properties for \(\mathrm{PD}(n)\). Later, \textit{E.X.W.~Xia} [J. Number Theory 159, 160--175 (2016; Zbl 1330.11066)] proved some infinite families of congruences modulo small powers of \(3\) for \(\mathrm{PD}(n)\). Recently, \textit{B.L.S.~Lin} [J. Number Theory 184, 216--234 (2018; Zbl 1375.05021)] introduced two new functions \(\mathrm{PD}_\mathrm{t}(n)\) and \(\mathrm{PDO}_\mathrm{t}(n)\), which denote the number of tags (designated summands) in the partitions enumerated, respectively, by \(\mathrm{PD}(n)\) and \(\mathrm{PDO}(n)\). He derived that \begin{align*} \sum_{n=0}^\infty\mathrm{PD}_\mathrm{t}(n)q^n &=\frac{1}{2}{\left(\frac{(q^3;q^3)_\infty^5} {(q;q)_\infty^3(q^6;q^6)_\infty^2}-\frac{(q^6;q^6)_\infty} {(q;q)_\infty(q^2;q^2)_\infty(q^3;q^3)_\infty}\right)} \end{align*} and \begin{align*} \sum_{n=0}^\infty\mathrm{PDO}_\mathrm{t}(n)q^n &=\frac{q(q^2;q^2)_\infty(q^3;q^3)_\infty^2 (q^{12};q^{12})_\infty^2}{(q;q)_\infty^2(q^6;q^6)_\infty}. \end{align*} Lin also obtained many congruences for \(\mathrm{PD}_\mathrm{t}(n)\) and \(\mathrm{PDO}_\mathrm{t}(n)\). For example, he proved that for any \(n\geq0\), \begin{align*} \mathrm{PDO}_\mathrm{t}(8n) &\equiv0\pmod{9},\tag{3}\\ \mathrm{PDO}_\mathrm{t}(24n) &\equiv0\pmod{27},\tag{4}\\ \mathrm{PDO}_\mathrm{t}(36n+24) &\equiv0\pmod{27}.\tag{5} \end{align*} The authors give some elementary proofs of some Ramanujan-type identities (including (1) and (2)) and congruences concerning \(\mathrm{PD}(n)\) and \(\mathrm{PDO}(n)\). Moreover, they improve on (3)--(5) by showing that \begin{align*} \mathrm{PDO}_\mathrm{t}(8n) &\equiv0\pmod{36},\\ \mathrm{PDO}_\mathrm{t}(24n) &\equiv0\pmod{3456},\\ \mathrm{PDO}_\mathrm{t}(36n+24) &\equiv0\pmod{3456}. \end{align*}