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

Aman Devnani; Pramod Eyyunni

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

T0 review · 0 major / 4 minor · reviewed 2026-05-10 · grok-4.3

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Pith's one-line read Distinct integer partitions can yield identical values under the pre_k maps derived from elementary symmetric polynomials, disproving a recent injectivity conjecture.

desk verdict The paper gives an explicit infinite family of counterexamples that disprove the injectivity conjecture for the pre_k maps on partitions, plus a revised conjecture and some alternate proofs for k=2 subcases. read the letter →

arxiv 2604.17424 v1 submitted 2026-04-19 math.CO math.NT

Aman Devnani , Pramod Eyyunni This is my paper · ORCID

classification math.COmath.NT

keywords integerpartitionselementarysymmetricpolynomialsinjectivitycounterexamplescombinatorialmapsfunctionsconjecturesincombinatorics

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

The paper constructs an infinite family of distinct integer partitions that map to the same output under each pre_k, where these maps come from applying the k-th elementary symmetric polynomial to the parts of a partition. A reader would care because this directly refutes the conjecture that the maps are injective, meaning different partitions would always produce different outputs. The authors respond by stating a modified version of the conjecture and by examining the relationships among the various pre_k maps instead of treating them separately. They also supply alternate proofs for three subcases of the k=2 map, which had already been settled elsewhere, and establish lower bounds on how many partitions of n lie in the image of pre_2.

What carries the argument

The pre_k maps on integer partitions, each tied to the k-th elementary symmetric polynomial applied to the parts of the partition.

What would settle it

Direct computation of the k-th elementary symmetric sums for the pairs or families of partitions given in the paper to check whether the outputs truly coincide while the partitions remain distinct.

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Extended reading notes

Core claim

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.

Load-bearing premise

The explicitly constructed distinct partitions produce exactly the same pre_k output under the chosen definition of the elementary symmetric polynomial maps.

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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.

Desk Editor's Note

read the letter

The main point is that the authors produce an infinite parametric family of distinct partitions that collide under the pre_k maps, which are built from elementary symmetric polynomials. This directly falsifies the conjecture from Ballantine and collaborators. They also state a modified version of the conjecture and supply alternate proofs for three subcases of the already-settled k=2 result, along with some lower bounds on the size of the image of pre_2. They look at how the maps for different k relate to one another instead of treating each one separately, which is a small shift in approach from the earlier papers they cite. The constructions are given explicitly enough that the collisions can be checked by direct substitution, and the stress-test note indicates the supporting arguments for the k=2 cases hold without obvious gaps. That counts as real progress on a narrow but concrete question. The modified conjecture is simply announced without supporting calculations or partial results, so it functions more as a suggestion than a developed claim. The lower-bound discussion for the image of pre_2 stays at a basic level and does not appear to improve on what follows from the counterexample construction itself. Overall the work stays inside the existing literature on partition maps and does not introduce new machinery or broader connections. Readers who track recent conjectures in partition combinatorics or who care about explicit counterexamples to injectivity statements will find the family useful to have on record. The paper is narrow enough that most people outside that subfield will not need it, but the central disproof is verifiable and corrects a specific recent claim, so it merits a serious referee who can check the parametric examples and the alternate proofs. I would send it to peer review rather than desk-reject it.

Referee Report

0 major / 4 minor

Summary. The paper disproves a conjecture of Ballantine et al. on the injectivity of the family of maps pre_k (k ≥ 1) from integer partitions to sequences of elementary symmetric polynomials by exhibiting an explicit infinite parametric family of distinct partitions λ ≠ μ such that pre_k(λ) = pre_k(μ) for each fixed k. It proposes a modified injectivity conjecture, develops relations among the pre_k maps for varying k, supplies three alternate proofs for subcases of the already-settled k = 2 conjecture, and derives lower bounds on the size of the image of pre_2 restricted to partitions of n.

Significance. The explicit infinite family of collisions supplies a definitive negative answer to the original conjecture and supplies concrete data for studying the modified version. The inter-map relations constitute a departure from the prior one-map-at-a-time literature. The alternate k = 2 proofs and the image-size lower bounds are useful additions even though the k = 2 case is independently settled. The work is therefore a solid contribution to the combinatorial study of symmetric-polynomial maps on partitions.

minor comments (4)

The precise definition of the map pre_k (how the k-th elementary symmetric polynomial is applied to the multiset of parts, and how the output sequence is indexed) should be stated with full notation in the introduction before any counterexample is presented.
In the section presenting the infinite family, include a short table or explicit numerical example for small k (e.g., k = 3) showing two distinct partitions and their common pre_k image; this would make the collision immediately verifiable.
The statement of the modified conjecture should be displayed as a numbered conjecture with the exact mathematical condition on the partitions.
The lower-bound argument for |im(pre_2) ∩ P(n)| should cite the exact combinatorial construction or injection used to obtain the bound.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of our manuscript, their assessment of its significance, and the recommendation for minor revision. The report correctly identifies the infinite family of counterexamples, the modified conjecture, the inter-map relations, the alternate proofs for the k=2 case, and the image-size bounds as the main contributions.

Circularity Check

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No significant circularity

full rationale

The paper disproves the Ballantine et al. conjecture on injectivity of pre_k maps by supplying an explicit parametric family of distinct partitions that collide under the map (defined via elementary symmetric polynomials). It also supplies alternate proofs for three k=2 subcases and lower bounds on the size of the image of pre_2. All load-bearing steps are direct combinatorial constructions and verifications; none reduce by definition, by fitting, or by a self-citation chain to the target claim itself. The work is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The claims rest on standard definitions of integer partitions and elementary symmetric polynomials together with explicit combinatorial constructions; no free parameters or new entities are introduced in the abstract.

assumptions (2)

standard math Integer partitions are finite non-increasing sequences of positive integers.
This is the standard definition invoked when applying symmetric polynomials to partitions.
standard math Elementary symmetric polynomials are the standard sums of products of distinct variables taken k at a time.
The maps pre_k are defined by applying these polynomials to the parts of a partition.

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Cite this review

Pith. "Pith review of Elementary symmetric polynomials and a potentially injective family of maps on partitions." pith.science (2026). https://pith.science/paper/2604.17424

@misc{pith2026260417424,
  author       = {Pith},
  title        = {Pith review of: Elementary symmetric polynomials and a potentially injective family of maps on partitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.17424}},
  note         = {Machine review of arXiv:2604.17424}
}

read the original 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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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Counterexamples regarding elementary symmetric partitions
math.CO 2026-05 unverdicted novelty 8.0 of 10

Disproves the refined conjecture that pre_j is injective for length > j by exhibiting non-injective cases at length 2j (j≥3), and proves the complete homogeneous map prh_j is injective on all partitions.
A note on partitions in the image of pre$_2$
math.CO 2026-06 unverdicted novelty 6.0 of 10

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 unverdicted novelty 5.0 of 10

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.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · cited by 3 Pith papers

[1]

Ballantine, G

C. Ballantine, G. Beck and M. Merca,Partitions and elementary symmetric polynomials: an experimental approach, Ramanujan J.,66(2)(2025), Paper No. 34

work page 2025
[2]

Ballantine, G

C. Ballantine, G. Beck, M. Merca and B. Sagan,Elementary symmetric partitions, Ann. Comb. (2024), online first

work page 2024
[3]

Ballantine, S

C. Ballantine, S. Nazir, B. E. Tenner, K. Westrem and C. Zhao,On partitions associated with elementary symmetric polynomials, Ramanujan J.,69(2)(2026), Paper No. 29

work page 2026
[4]

S. J. Li,A note on multiset reconstruction from pairwise products and total sum, Integers26(2026), Paper No. A16

work page 2026
[5]

Schneider and A

R. Schneider and A. Sills,The product of parts or “norm” of a partition, Integers20A(2020), Proceedings of the Integers Conference 2018, Paper No. A13. Aman Devnani, Birla Institute of Technology & Science Pilani, Vidya vihar, Pilani, Ra- jasthan - 333031, India. Email address:f20231026@pilani.bits-pilani.ac.in Pramod Eyyunni, Department of Mathematics, B...

work page 2020

Pith tools

Reviewed May 10, 2026 · model on record in the stance chip above.

[1] [1]

Ballantine, G

C. Ballantine, G. Beck and M. Merca,Partitions and elementary symmetric polynomials: an experimental approach, Ramanujan J.,66(2)(2025), Paper No. 34

work page 2025

[2] [2]

Ballantine, G

C. Ballantine, G. Beck, M. Merca and B. Sagan,Elementary symmetric partitions, Ann. Comb. (2024), online first

work page 2024

[3] [3]

Ballantine, S

C. Ballantine, S. Nazir, B. E. Tenner, K. Westrem and C. Zhao,On partitions associated with elementary symmetric polynomials, Ramanujan J.,69(2)(2026), Paper No. 29

work page 2026

[4] [4]

S. J. Li,A note on multiset reconstruction from pairwise products and total sum, Integers26(2026), Paper No. A16

work page 2026

[5] [5]

Schneider and A

R. Schneider and A. Sills,The product of parts or “norm” of a partition, Integers20A(2020), Proceedings of the Integers Conference 2018, Paper No. A13. Aman Devnani, Birla Institute of Technology & Science Pilani, Vidya vihar, Pilani, Ra- jasthan - 333031, India. Email address:f20231026@pilani.bits-pilani.ac.in Pramod Eyyunni, Department of Mathematics, B...

work page 2020