Remarks on $d$-ary partitions and an application to elementary symmetric partitions

We prove new formulas for $p_d(n)$, the number of $d$-ary partitions of $n$, and, also, for its polynomial part. Given a partition $\lambda =({\lambda }_1,\dots ,{\lambda }_{\mathrm{\ell }})$, its associated $j$-th symmetric elementary partition, $pr{e}_j(\lambda )$, is the partition whose parts are $\{{\lambda }_{i_1}\cdots {\lambda }_{i_j}:1\le i_1<\cdots <i_j\le \mathrm{\ell }\}$. We prove that if $\lambda$ and $\mu$ are two $d$-ary partitions of length $\mathrm{\ell }$ such that $pr{e}_j(\lambda )=pr{e}_j(\mu )$ and ${\lambda }_{i_1}\cdots {\lambda }_{i_j}={\mu }_{i_1}\cdots {\mu }_{i_j}$, for all $1\le i_1<\cdots <i_j\le \mathrm{\ell }$, then $\lambda =\mu$.