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Counterexamples regarding elementary symmetric partitions

T0 review · 0 major / 2 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read The pre_j map from partitions via the j-th elementary symmetric polynomial is not injective on those with exactly 2j parts for j at least 3.

desk verdict Gives explicit counterexamples killing the refined pre_j conjecture at length 2j for j>=3 and proves injectivity for the complete homogeneous map prh_j. read the letter →

arxiv 2606.00420 v3 pith:EBWY2J3T submitted 2026-05-29 math.CO

Vixail Hadelyn , Harper Niergarth , Weiyou Li , Wenhui Li This is my paper · ORCID
classification math.CO
keywords elementarysymmetricpolynomialspartitionmapsinjectivitycounterexamplescompletehomogeneousfunctionspartitions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Ballantine, Beck, and Merca defined the pre_j map that takes a partition and produces a new partition whose parts are the individual summands appearing when the j-th elementary symmetric polynomial is evaluated on the original parts. They conjectured that this map is injective among all partitions of any n that have length at least j. After the case of length exactly j was already shown to fail, a refined conjecture held that injectivity would hold whenever length exceeds j. The paper supplies explicit counterexamples proving the map is not injective when length equals 2j for every j of size 3 or larger. It additionally shows that the parallel map prh_j built from the complete homogeneous symmetric polynomial instead of the elementary one is injective across every partition.

What carries the argument

the pre_j map, which expands the j-th elementary symmetric polynomial on the parts of an input partition and collects the resulting summands into an output partition

What would settle it

An explicit pair of distinct partitions, each having length exactly 2j for some j ≥ 3, that produce identical outputs under pre_j.

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Extended reading notes

Core claim

The pre_j map is not injective on the set of partitions of n that have length exactly 2j, for each j ≥ 3; explicit counterexamples establish this failure of the refined conjecture. The map prh_j defined analogously with complete homogeneous symmetric polynomials is injective on the collection of all partitions.

Load-bearing premise

The pre_j map is applied exactly as originally defined, without modification, to the partitions of length 2j that serve as the counterexamples.

Editorial extensions

If this is right

  • pre_j fails to be injective on partitions of length 2j for every j ≥ 3
  • the refined conjecture that pre_j is injective for all lengths strictly greater than j is false
  • prh_j is injective on the set of every partition

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Collisions under pre_j at length 2j suggest that similar non-injectivity may appear at other lengths that are multiples of j.
  • The contrast between pre_j and the always-injective prh_j indicates that the choice between elementary and complete homogeneous polynomials controls whether the induced map on partitions preserves distinct inputs.
  • For small values such as j = 3 the paper's counterexamples could be used to locate the smallest n at which the first collision occurs.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 2 minor

Summary. The paper disproves a refined conjecture of Devnani and Eyyunni by exhibiting explicit counterexamples showing that the elementary symmetric partition map pre_j is not injective on partitions of n having length exactly 2j when j ≥ 3. It further proves that the analogous map prh_j, defined via the complete homogeneous symmetric polynomial, is injective on the set of all partitions.

Significance. The explicit constructions settle the refined conjecture in the negative for the indicated length and supply a positive injectivity theorem for the homogeneous variant. These results clarify the range of injectivity for symmetric-polynomial-induced maps on partitions and furnish concrete examples that can be used to test further conjectures in the area.

minor comments (2)
  1. [Introduction / Main results] The statement of the main theorem (presumably Theorem 1.1 or 3.1) would benefit from an explicit small-j example (e.g., j=3) placed immediately after the general construction so that the length-2j condition and the image coincidence can be verified by direct inspection.
  2. [Section on prh_j injectivity] In the proof that prh_j is injective, the recovery argument from the multiset of values back to the original partition should be cross-referenced to the precise definition of the complete homogeneous polynomial evaluation used in the map.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, accurate summary of the results, and recommendation to accept the manuscript. The report correctly identifies the counterexamples disproving the refined conjecture for pre_j on partitions of length 2j (j ≥ 3) and the injectivity theorem for prh_j.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The paper disproves the refined conjecture via explicit constructions showing non-injectivity of pre_j on length-2j partitions for j≥3, together with a separate proof that prh_j is injective on all partitions. These are direct mathematical arguments resting on the standard definition of the maps (applied without modification to the constructed examples) and standard partition theory; no equations reduce by construction to fitted inputs, no load-bearing self-citations form the central claim, and no ansatz or uniqueness result is smuggled in. The derivation is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The work rests on the standard combinatorial definitions of partitions and symmetric polynomials introduced in the cited papers; no new entities or fitted parameters are introduced.

assumptions (1)
  • standard math Standard algebraic properties of elementary and complete homogeneous symmetric polynomials on finite sets of positive integers
    Invoked in the definition of pre_j and prh_j.

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Cite this review

Pith. "Pith review of Counterexamples regarding elementary symmetric partitions." pith.science (2026). https://pith.science/paper/EBWY2J3T

@misc{pith2026260600420,
  author       = {Pith},
  title        = {Pith review of: Counterexamples regarding elementary symmetric partitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EBWY2J3T}},
  note         = {Machine review of arXiv:2606.00420}
}
abstract

Ballantine, Beck, and Merca defined the elementary symmetric partition map pre$_j$ that sends a partition $\lambda$ to a larger partition whose parts are the summands appearing in the evaluation of the $j$-th elementary symmetric polynomial on $\lambda$. 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.

Figures

Figures reproduced from arXiv: 2606.00420 by the authors.

Figure 1
Figure 1. Examples of Q-embeddings. Example 3.6. Consider the embeddings G and H appearing on the left and right in [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. An example of a Q-embedding not satisfying (P2). In this article, we do not explicitly classify which embeddings satisfy (P2). Instead, we will show that if G is an embedding that fails to satisfy (P2), we may “wiggle” the vertices of G to obtain a new embedding G′ that does. In fact we will be able to show this for U-embeddings for arbitrary dense sets U. In particular, we may set U = Q which lets us find rational … view at source ↗

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Works this paper leans on

9 extracted references · 1 canonical work pages

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    Ian Grant Macdonald,Symmetric functions and hall polynomials, Oxford university press, 1998. 18 V. HADELYN, H. NIERGARTH, W. LI, W. LI

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  1. [10]

    University of W aterloo, Ontario Canada

    Rohith Thomas and Katherine Tung,Injectivity of symmetric polynomial maps on parti- tions, arXiv preprint arXiv:2604.17424 (2026). University of W aterloo, Ontario Canada. Email address:lynHadelyn@gmail.com University of W aterloo, Ontario Canada. Email address:hniergarth@uwat...

    2026 arXiv

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