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| Autori principali: | , , , |
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| Natura: | Preprint |
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2026
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| Accesso online: | https://arxiv.org/abs/2606.00420 |
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| _version_ | 1866910273701412864 |
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| author | Hadelyn, Vixail Niergarth, Harper Li, Weiyou Li, Wenhui |
| author_facet | Hadelyn, Vixail Niergarth, Harper Li, Weiyou Li, Wenhui |
| contents | Ballantine, Beck, and Merca defined the elementary symmetric partition map pre$_j$ that sends a partition $λ$ to a larger partition whose parts are the summands appearing in the evaluation of the $j$-th elementary symmetric polynomial on $λ$. They conjectured that pre$_j$ is injective on the set of partitions of $n$ with length $\ell \geq j$. The $\ell = j$ case was disproved by Devnani and Eyyunni; they instead conjectured the statement to be true for $\ell > j$. In this article, we answer this refined conjecture in the negative by proving that pre$_j$ is not injective on partitions of $n$ with length $2j$ for $j \geq 3$. We also prove that the analogous map prh$_j$ defined via the complete homogenous symmetric polynomial is injective on the set of all partitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_00420 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Counterexamples regarding elementary symmetric partitions Hadelyn, Vixail Niergarth, Harper Li, Weiyou Li, Wenhui Combinatorics Ballantine, Beck, and Merca defined the elementary symmetric partition map pre$_j$ that sends a partition $λ$ to a larger partition whose parts are the summands appearing in the evaluation of the $j$-th elementary symmetric polynomial on $λ$. They conjectured that pre$_j$ is injective on the set of partitions of $n$ with length $\ell \geq j$. The $\ell = j$ case was disproved by Devnani and Eyyunni; they instead conjectured the statement to be true for $\ell > j$. In this article, we answer this refined conjecture in the negative by proving that pre$_j$ is not injective on partitions of $n$ with length $2j$ for $j \geq 3$. We also prove that the analogous map prh$_j$ defined via the complete homogenous symmetric polynomial is injective on the set of all partitions. |
| title | Counterexamples regarding elementary symmetric partitions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2606.00420 |