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

TL;DR

This paper derives new formulas for counting $d$-ary partitions of integers and explores properties of elementary symmetric partitions, establishing conditions under which two $d$-ary partitions are identical based on their symmetric elementary partitions.

Contribution

It introduces new formulas for $p_d(n)$ and characterizes when two $d$-ary partitions are equal using elementary symmetric partitions.

Findings

New formulas for $p_d(n)$ and its polynomial part.

A uniqueness condition for $d$-ary partitions based on symmetric elementary partitions.

Insights into the structure of $d$-ary partitions.

Abstract

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 $\lambda=\mu$.