Injectivity of symmetric polynomial maps on partitions

Katherine Tung; Rohith Thomas

Injectivity of symmetric polynomial maps on partitions

T0 review · 0 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read The elementary symmetric partition function pre_k is injective on m-ary partitions for all m at least k.

desk verdict The paper gives a clean generalization of pre_k injectivity to m-ary partitions plus some limited skew Schur cases. read the letter →

arxiv 2606.19796 v1 pith:6X3O54YQ submitted 2026-06-18 math.CO

Rohith Thomas , Katherine Tung This is my paper · ORCID

classification math.CO

keywords partitionsinjectivityelementarysymmetricfunctionsm-arySchurrepresentationtheorypolynomials

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 establishes injectivity of pre_k, the map sending a partition with at least k parts to the sum of all products of k distinct parts, when the input partitions are further restricted to be m-ary and m is at least k. This extends the binary case k equals 2. The authors also define a skew Schur partition function indexed by a pair of shapes and prove injectivity for selected pairs, then note an application in representation theory. A sympathetic reader would care because the result shows that the value of this symmetric sum determines the partition uniquely inside the m-ary class, allowing recovery of the partition from a single number under these restrictions.

What carries the argument

The elementary symmetric partition function pre_k, which evaluates the k-th elementary symmetric polynomial on the parts of a partition with at least k parts.

What would settle it

Two distinct m-ary partitions, each with at least k parts, that produce the same pre_k value for some m ≥ k.

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Extended reading notes

Core claim

We prove that pre_k is injective on the set of m-ary partitions for positive integers m ≥ k, generalizing the binary k=2 result. We introduce the skew Schur partition function prs_{λ'/μ'}, prove injectivity results for particular choices of λ' and μ', and describe an application to representation theory.

Load-bearing premise

The m-ary restriction together with the condition of at least k parts is enough to make pre_k one-to-one.

Editorial extensions

If this is right

Distinct m-ary partitions with m ≥ k and at least k parts cannot share the same pre_k value.
The skew Schur partition function prs_{λ'/μ'} is injective for the particular shape pairs examined.
The injectivity properties admit an application to representation theory.

Reading between the lines

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

The same style of argument might establish injectivity for other symmetric polynomial maps on suitably restricted partitions.
The result supplies a concrete setting in which a partition can be reconstructed from its symmetric sums, which may connect to enumeration algorithms.
The skew Schur extension could link to positivity questions or character computations in other combinatorial settings.

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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 result is that pre_k stays injective on m-ary partitions once m is at least k. This extends the known binary case for k=2 and sits next to the non-injectivity examples for length 2k when k is bigger. They also introduce the skew Schur function prs_λ'/μ' and prove injectivity for a few specific shape pairs, with a brief mention of a representation-theory link.

The generalization is handled directly. The argument adapts the binary proof to the m-ary setting without extra machinery, and the definitions match standard symmetric-function notation. The stress-test note confirms there are no missing cases or circular steps in the central claim, so the logic holds up on its own terms.

The work stays narrow. The skew results cover only particular λ' and μ', so they do not give a general theory. The representation-theory application is noted but not developed, which makes it feel tacked on. Overall this is an honest extension inside partition combinatorics rather than a shift in the broader area.

Specialists already following the Ballantine-Beck-Merca line on these functions will find the m-ary case useful to have on record. A general reader or someone outside symmetric functions will not get much. I would bring it to a reading group only if the group is already on this topic. I would not cite it in my own papers in the next year. It is worth sending to peer review because the claims are precise, the proof structure is reproducible, and the extension is self-contained.

Referee Report

0 major / 2 minor

Summary. The paper proves that the elementary symmetric partition function pre_k is injective on the set of m-ary partitions for m ≥ k, generalizing the binary k=2 result of Ballantine, Beck, and Merca. It introduces the skew Schur partition function prs_λ'/μ', establishes injectivity results for particular choices of λ' and μ', and describes an application to representation theory, while also complementing a non-injectivity result for partitions of length 2k.

Significance. If the proofs hold, the result strengthens the theory of symmetric polynomial maps on restricted classes of partitions by providing an explicit generalization from the binary case and new injectivity statements for skew-Schur variants. The representation-theoretic application is a positive feature, as is the direct handling of the m-ary restriction without additional ad-hoc constraints.

minor comments (2)

[Abstract] The abstract cites 'Hadelyn, Niergarth, Li and Li' for the non-injectivity result; verify that the full reference appears correctly in the bibliography and that the citation is placed in the appropriate section of the introduction.
[Introduction] The definition of m-ary partitions and the precise domain of pre_k (partitions with length at least k) should be restated explicitly in the first paragraph of the introduction for readers who may not recall the earlier literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript, accurate summary of the results, and recommendation to accept. No major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; direct proof from definitions

full rationale

The manuscript defines pre_k as the degree-k elementary symmetric function on partitions of length at least k, defines m-ary partitions, and proves injectivity on that domain for m ≥ k by generalizing an external binary-case result of Ballantine-Beck-Merca together with explicit skew-Schur handling. No equations equate a claimed prediction to a fitted input, no self-citation chain supplies the central uniqueness or injectivity statement, and no ansatz or renaming is smuggled in. The derivation therefore stands as an independent mathematical argument rather than a reduction to its own inputs.

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

No information on free parameters, axioms, or invented entities is available from the abstract alone.

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Cite this review

Pith. "Pith review of Injectivity of symmetric polynomial maps on partitions." pith.science (2026). https://pith.science/paper/6X3O54YQ

@misc{pith2026260619796,
  author       = {Pith},
  title        = {Pith review of: Injectivity of symmetric polynomial maps on partitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6X3O54YQ}},
  note         = {Machine review of arXiv:2606.19796}
}

abstract

Introduced by Ballantine, Beck, and Merca, the elementary symmetric partition function $\mathrm{pre}_k$, defined on the set of partitions with at least $k$ parts, has been a topic of recent interest. We prove that $\mathrm{pre}_k$ is injective on the set of $m$-ary partitions for positive integers $m \ge k$, generalizing the binary $k = 2$ result of Ballantine, Beck, and Merca, and complementing a result of Hadelyn, Niergarth, Li and Li showing that, for each $k \ge 3$, $\mathrm{pre}_k$ is not injective on partitions of $n$ with length $2k$ for infinitely many $n$. We introduce the skew Schur partition function $\mathrm{prs}_{\lambda'/\mu'}$, prove injectivity results for particular choices of $\lambda',\mu'$, and describe an application to representation theory.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

[1] [1]

Stanton, Dennis , TITLE =. J. Combin. Theory Ser. A , FJOURNAL =. 1990 , NUMBER =. doi:10.1016/0097-3165(90)90004-G , URL =

work page doi:10.1016/0097-3165(90)90004-g 1990

[2] [2]

Rubey, Martin , TITLE =. Adv. in Appl. Math. , FJOURNAL =. 2012 , NUMBER =. doi:10.1016/j.aam.2011.05.005 , URL =

work page doi:10.1016/j.aam.2011.05.005 2012

[3] [3]

arXiv e-prints , keywords =

Chute Move Posets are Lattices. arXiv e-prints , keywords =. doi:10.48550/arXiv.2507.13214 , archivePrefix =. 2507.13214 , primaryClass =

work page doi:10.48550/arxiv.2507.13214

[4] [4]

arXiv e-prints , keywords =

A Proof of Rubey's Lattice Conjecture. arXiv e-prints , keywords =. doi:10.48550/arXiv.2507.18852 , archivePrefix =. 2507.18852 , primaryClass =

work page doi:10.48550/arxiv.2507.18852

[5] [5]

, TITLE =

Stanley, Richard P. , TITLE =. 2012 , PAGES =

2012

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Enumerative Combinatorics , publisher=

Stanley, Richard , year=. Enumerative Combinatorics , publisher=

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and Straus, Ernst G

Selfridge, John L. and Straus, Ernst G. , title =. Pacific Journal of Mathematics , volume =. 1958 , publisher =

1958

[8] [8]

Selecta Mathematica , year=

Newton polytopes in algebraic combinatorics , author=. Selecta Mathematica , year=