Remarks on $d$-ary partitions and an application to elementary symmetric partitions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909990081527808 |
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| author | Cimpoeas, Mircea Tanase, Roxana |
| author_facet | Cimpoeas, Mircea Tanase, Roxana |
| contents | 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 $λ=μ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_04459 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Remarks on $d$-ary partitions and an application to elementary symmetric partitions Cimpoeas, Mircea Tanase, Roxana Combinatorics 11P81, 11P83 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 $λ=μ$. |
| title | Remarks on $d$-ary partitions and an application to elementary symmetric partitions |
| topic | Combinatorics 11P81, 11P83 |
| url | https://arxiv.org/abs/2506.04459 |