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 $\lambda=(\lambda_1,\ldots,\lambda_{\ell})$, its associated $j$-th symmetric elementary partition, $pre_{j}(\lambda)$, is the partition whose parts are $\{\lambda_{i_1}\cdots\lambda_{i_j}\;:\;1\leq i_1 < \cdots < i_j\leq \ell\}$. We prove that if $\lambda$ and $\mu$ are two $d$-ary partitions of length $\ell$ such that $pre_j(\lambda)=pre_j(\mu)$ and $\lambda_{i_1}\cdots \lambda_{i_j} = \mu_{i_1}\cdots \mu_{i_j}$, for all $1\leq i_1 < \cdots < i_j\leq \ell$, then $\lambda=\mu$.