REVIEW 1 major objections 1 minor 2 references
A note on partitions in the image of pre$_2$
T0 review · 1 major / 1 minor · reviewed 2026-06-28 · grok-4.3
Pith's one-line read Exactly one partition of n lies in the image of pre₂ only for n in {1, 2, 4}.
desk verdict This note cleanly resolves the open question by pinning down exactly when the image of pre₂ has size one. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The pre₂ map obtained by applying the second elementary symmetric polynomial to the parts of an integer partition.
What would settle it
Discovery of any n ≥ 5 whose image under pre₂ contains exactly one partition of n.
Extended reading notes
Core claim
The only values of n for which exactly one partition of n lies in the image of pre₂ are 1, 2 and 4. For every n ≥ 5 at least two partitions of n lie in the image of pre₂.
Load-bearing premise
The enumeration of partitions in the image of pre₂ for each n is complete and free of omissions.
Editorial extensions
If this is right
- Exactly one partition of 1 is in the image of pre₂.
- Exactly one partition of 2 is in the image of pre₂.
- Exactly one partition of 4 is in the image of pre₂.
- At least two partitions of every n ≥ 5 are in the image of pre₂.
Reading between the lines
- The prior question of Devnani and Eyyunni is now settled for k=2.
- The same multiplicity question for pre_k with k > 2 remains open.
- The explicit constructions used for n ≥ 5 supply concrete examples of multiple realizations under the map.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript resolves a question posed by Devnani and Eyyunni on the pre₂ map (arising from the second elementary symmetric polynomial on partition parts). It claims that exactly one partition of n lies in the image of pre₂ precisely when n ∈ {1, 2, 4}, and that for every n ≥ 5 the image contains at least two distinct partitions of n.
Significance. If the proof is correct, the result gives a complete and sharp answer to the existence question, identifying all exceptional n and establishing a uniform lower bound of two for larger n. This is a modest but clean contribution to the study of these symmetric-polynomial maps on partitions.
major comments (1)
- [argument for n ≥ 5] The load-bearing step is the argument establishing that the image has cardinality at least 2 for every n ≥ 5. The manuscript must supply either an exhaustive check up to a sufficient bound together with a uniform construction that always produces two or more distinct outputs, or a case analysis whose completeness is explicitly verified; any omitted n in this range would falsify the universal claim.
minor comments (1)
- [Introduction] The abstract and introduction should cite the precise statement of the question from Devnani–Eyyunni (including the definition of pre₂) so that the contribution is self-contained.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for identifying the need to strengthen the presentation of the argument for n ≥ 5. We address the major comment below.
read point-by-point responses
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Referee: [argument for n ≥ 5] The load-bearing step is the argument establishing that the image has cardinality at least 2 for every n ≥ 5. The manuscript must supply either an exhaustive check up to a sufficient bound together with a uniform construction that always produces two or more distinct outputs, or a case analysis whose completeness is explicitly verified; any omitted n in this range would falsify the universal claim.
Authors: The manuscript already contains a uniform construction that produces two distinct partitions in the image of pre₂ for every n ≥ 5. The construction proceeds by exhibiting two families of partitions (one with a distinguished part of size 2 and another with two parts of size 1, adjusted by the remaining mass) whose images under pre₂ are distinct yet both sum to n; the algebraic verification that these images differ holds identically for all n ≥ 5 and does not rely on case distinctions. To make the completeness explicit as requested, we will add a short table verifying the construction for 5 ≤ n ≤ 12 together with a sentence confirming that the same formulas apply without modification for all larger n. This supplies both the uniform construction and a finite explicit check. revision: partial
Circularity Check
No circularity; proof is self-contained via external definition and direct argument.
full rationale
The paper adopts the definition and basic properties of the pre₂ map from the cited Devnani–Eyyunni work without re-derivation and then supplies an explicit proof that the image cardinality is 1 only for n in {1,2,4} and at least 2 for all n≥5. No load-bearing step reduces by construction to a fitted parameter, a self-referential definition, or a self-citation chain; the argument consists of case analysis or uniform construction whose validity is independent of the target statement. This is the normal situation for a short note in combinatorics.
Assumptions & free parameters
assumptions (1)
- domain assumption The pre₂ map is obtained by applying the second elementary symmetric polynomial to the parts of an integer partition.
Cite this review
Pith. "Pith review of A note on partitions in the image of pre$_2$." pith.science (2026). https://pith.science/paper/QW6M2MKR
@misc{pith2026260602683,
author = {Pith},
title = {Pith review of: A note on partitions in the image of pre$_2$},
year = {2026},
howpublished = {\url{https://pith.science/paper/QW6M2MKR}},
note = {Machine review of arXiv:2606.02683}
}
abstract
Devnani and Eyyunni recently studied the maps pre$_k$ on integer partitions, which arise from applying elementary symmetric polynomials to the parts of a partition. They asked whether there exists $n \ge 1$ such that exactly one partition of $n$ lies in the image of pre$_2$. We show that this occurs only for $n$ in {1, 2, 4}, and that for all $n \ge 5$, at least two partitions of n are in the image of pre$_2$.
Reference graph
Works this paper leans on
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[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
2025
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[2]
Elementary symmetric polynomials and a potentially injective family of maps on partitions
A. Devnani and P. Eyyunni,Elementary symmetric polynomials and a potentially injective family of maps on partitions, arXiv:2604.17424, 2026. 2
work page Pith review arXiv 2026
Reviewed June 28, 2026 · model on record in the stance chip above.
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