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 $λ=(λ_1,\ldots,λ_{\ell})$, its associated $j$-th symmetric elementary partition, $pre_{j}(λ)$, is the partition whose parts are $\{λ_{i_1}\cdotsλ_{i_j}\;:\;1\leq i_1 < \cdots < i_j\leq \ell\}$. We prove that if $λ$ and $μ$ are two $d$-ary partitions of length $\ell$ such that $pre_j(λ)=pre_j(μ)$ and $λ_{i_1}\cdots λ_{i_j} = μ_{i_1}\cdots μ_{i_j}$, for all $1\leq i_1 < \cdots < i_j\leq \ell$, then $λ=μ$.