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

TL;DR

This paper advances the theory of $d$-ary partitions by establishing a length-preserving bijection between ordinary partitions and $d$-ary partitions via $ ext{Exp}_d$ and $ ext{Log}_d$, enabling explicit finite-sum formulas for $p_d(n)$ and its polynomial part $P_d(n)$ through Sylvester wave decomposition with $k=loor{d{ot}d{log}_d(n)}$ and $D=d^k$. It provides several equivalent closed forms, including a Bernoulli-number expansion for $P_d(n)$, and demonstrates practical calculations such as $p_3(8)=3$ and $p_3(20)=12$. As an application, it analyzes elementary symmetric partitions, proving a uniqueness result: if $ ext{pre}_j(oldsymbol ymbda)= ext{pre}_j(oldsymbol ymu)$ and $oldsymbol ymbla_{i_1}oldsymbol ymbla_{i_j}=oldsymbol ymu_{i_1}oldsymbol ymbmu_{i_j}$ for all index tuples, then $oldsymbol ymbda=oldsymbol ymu$, via a determinant argument for a circulant-like matrix with $ ext{det}(C)=j$. These results connect partition counting with explicit arithmetic formulas and linear-algebraic techniques, broadening computational and structural understanding of $d$-ary partitions and their symmetric counterparts.

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 $λ=(λ_1,\ldots,λ_{\ell})$, 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 $λ=μ$.

Theorems & Definitions (23)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Definition 3.1
• Definition 3.2
• Proposition 3.3
• proof
• Lemma 3.4
• proof