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Elementary symmetric polynomials and a potentially injective family of maps on partitions

· 2026 · math.CO · arXiv 2604.17424

3 Pith papers cite this work. Polarity classification is still indexing.

3 Pith papers citing it
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abstract

In this article, we provide an infinite family of examples to disprove a recent conjecture due to Ballantine and her collaborators on the injectivity of a class of maps, namely pre_k, defined on integer partitions. These maps arise from applying the sequence of elementary symmetric polynomials to integer partitions, where pre_k is associated with the kth polynomial. Subsequently, we state a modified version of their conjecture. Throwing fresh light on these class of maps, we study the inter-relationships between them, deviating from the approaches so far, which study these maps one at a time. Though one case of the conjecture (k=2) has now been settled independently by the work of Ballantine and collaborators, and Li, we provide alternate proofs of three subcases corresponding to this settled case. We also discuss lower bounds for the number of partitions of n which are in the image of the map pre_2.

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math.CO 3

years

2026 3

verdicts

UNVERDICTED 3

representative citing papers

A note on partitions in the image of pre$_2$

math.CO · 2026-06-01 · unverdicted · novelty 6.0

The paper shows that the image of pre₂ contains exactly one partition of n only for n=1,2,4 and at least two partitions for all n≥5.

Injectivity of symmetric polynomial maps on partitions

math.CO · 2026-06-18 · unverdicted · novelty 5.0

Proves injectivity of pre_k on m-ary partitions for m≥k and introduces skew Schur partition function prs_λ'/μ' with injectivity results for particular choices.

citing papers explorer

Showing 3 of 3 citing papers.

  • A note on partitions in the image of pre$_2$ math.CO · 2026-06-01 · unverdicted · none · ref 2 · internal anchor

    The paper shows that the image of pre₂ contains exactly one partition of n only for n=1,2,4 and at least two partitions for all n≥5.

  • Counterexamples regarding elementary symmetric partitions math.CO · 2026-05-29 · unverdicted · none · ref 5 · 2 links · internal anchor

    Counterexamples prove pre_j is not injective on partitions of length 2j for j≥3, while the complete homogeneous map prh_j is injective on all partitions.

  • Injectivity of symmetric polynomial maps on partitions math.CO · 2026-06-18 · unverdicted · none · ref 21 · internal anchor

    Proves injectivity of pre_k on m-ary partitions for m≥k and introduces skew Schur partition function prs_λ'/μ' with injectivity results for particular choices.