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REVIEW 3 major objections 3 minor 22 references

Chromatic MacMahon symmetric functions of graphs

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The chromatic symmetric MacMahon function of a vertex-weighted tree determines the generating function that counts every vertex subset by cardinality, total weight, and numbers of internal and external edges.

desk verdict A clean, natural generalization of the Crew-conjecture theorem to vertex-weighted trees via a two-alphabet MacMahon invariant; the proof is uncheckable in the copy I have, but the claim is plausible and deserves a referee. read the letter →

arxiv 2508.00157 v1 pith:FG5Y5PFY submitted 2025-07-31 math.CO

Jeremy L. Martin , May B. Trist This is my paper · ORCID
classification math.CO MSC 05E0505C1505C0505A15
keywords chromaticsymmetricfunctionMacMahonvertex-weightedgraphspropercoloringstreesgeneratingfunctionsgroupaction
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

This paper introduces the chromatic symmetric MacMahon function, a two-alphabet invariant of vertex-weighted graphs that records, for every proper coloring, both the sizes and the weights of the color classes. The paper's central claim is that for a tree this single invariant determines the generating function that enumerates all vertex subsets by cardinality, total weight, number of internal edges, and number of external edges. If the claim is correct, the new invariant carries the extra vertex-weight information that the ordinary chromatic symmetric function ignores, making it a strictly richer object for weighted trees. The theorem also extends the previously proved unweighted tree result, showing that the weighted case is a genuine generalization rather than a separate problem.

What carries the argument

The carrying object is the chromatic symmetric MacMahon function itself: a formal power series in two alphabets of variables, invariant under the symmetric group acting diagonally on the two alphabets, obtained by summing over all proper colorings of the vertex-weighted graph a monomial per color class that records the class's size in the first alphabet and its total vertex weight in the second. The key structural fact used by the proof is that this two-alphabet encoding keeps cardinality data and weight data in separate coefficients, so the coefficient of each monomial in the first alphabet is a polynomial in the second alphabet. The theorem is the statement that for trees this coefficient structure is exactly sharp enough to recover the weighted subset generating function with its internal and external edge counts.

What would settle it

Compute the chromatic symmetric MacMahon function for all vertex-weighted trees on, say, seven vertices with generic, formally independent vertex weights, and compare it with the weighted subset generating function; if two trees share the invariant but have different subset generating functions, the theorem is refuted.

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Extended reading notes

Core claim

The central discovery is that the chromatic symmetric MacMahon function of a tree is a complete package for the tree's weighted subset census. Concretely, the paper proves that from the invariant $\Psi_T(\mathbf{x};\mathbf{y})$, which sums over proper colorings a monomial recording each color class's size in one alphabet and its total vertex weight in the other, one can recover the enumerator $\sum_{S\subseteq V(T)} u^{|S|} t^{\mathrm{wt}(S)} p^{e_{\mathrm{int}}(S)} q^{e_{\mathrm{ext}}(S)}$, where $\mathrm{wt}(S)$ is the total weight of $S$, $e_{\mathrm{int}}(S)$ counts edges with both endpoints in $S$, and $e_{\mathrm{ext}}(S)$ counts edges crossing the cut $(S,V(T)\setminus S)$. The two alphabets play complementary roles, one carrying color-class cardinalities and the other carrying vertex weights, and for trees no information is lost in the passage from colorings to subsets. This generalizes the unweighted theorem that the ordinary chromatic symmetric function of a tree determines its vertex-subset enumeration data.

Load-bearing premise

The proof rests on the assumption that the two sets of variables in the invariant remain truly independent, so that distinct weighted colorings cannot collapse to the same series and the weight information can still be read off coefficient by coefficient.

Editorial extensions

If this is right

  • Two vertex-weighted trees with different weighted subset generating functions must have different chromatic symmetric MacMahon functions, so the invariant separates any pair of trees that the subset census separates.
  • Setting all vertex weights equal recovers the unweighted theorem for trees: the MacMahon function reduces to the ordinary chromatic symmetric function and the subset census reduces to the unweighted subtree information.
  • For a single tree, the entire collection of subset data, including size, weight, internal edges, and external edges, can in principle be extracted from one two-alphabet series without listing all $2^n$ subsets separately.
  • The weighted generalization places the weighted tree case on the same footing as the unweighted tree case, so the remaining open territory for this style of invariant lies in graphs with cycles.

Reading between the lines

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

  • A natural next question, not settled by this paper, is whether generically chosen vertex weights make the weighted subset generating function itself a complete invariant of the tree; the two-alphabet invariant constructed here is the natural tool for testing that.
  • Because the subset census records internal and external edge counts for every subset, the invariant sits close to Tutte-polynomial-style data, and an explicit specialization connecting the MacMahon function to a weighted Tutte polynomial would be a natural next step.
  • One could compute the invariant for all small vertex-weighted trees with independent formal weights and check empirically whether the map from weighted trees to two-alphabet series is injective; the theorem guarantees at least the one-way determination proved here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper introduces a chromatic symmetric MacMahon function for vertex-weighted graphs, a two-alphabet symmetric function invariant of the diagonal action, and claims that for trees this invariant determines the generating function for vertex subsets by cardinality, weight, and the numbers of internal and external edges. The result is presented as a generalization of the unweighted Crew-conjecture theorem proved by Aliste-Prieto--Martin--Wagner--Zamora and Liu--Tang.

Significance. If the main theorem is correct, it extends a significant line of research on chromatic symmetric functions from unweighted to vertex-weighted graphs, with a genuinely new two-alphabet invariant. The claimed implication is strong: it would encode both the cardinality and the weight profile of every vertex subset along with its internal and cut edge counts, making the invariant at least as informative for weighted trees as the ordinary chromatic symmetric function is for unweighted trees. The construction is natural and the abstract is clean, but the supplied full text is entirely unreadable, so the proof cannot currently be checked. The significance is therefore conditional on a verifiable manuscript.

major comments (3)
  1. [Full text (entire manuscript)] The supplied full text is a corrupted encoding (mojibake) with no readable section, equation, or argument. I cannot verify the proof of the central theorem, nor the definitions and lemmas that would support it. This is load-bearing because the claimed two-alphabet independence and the extraction of the weighted subset generating function are precisely the steps that require careful checking. The authors must resubmit a clean, readable version before the paper can be evaluated.
  2. [Abstract and Theorem statement] The abstract does not specify the nature of the vertex weights: are they formal variables, generic values, or arbitrary commutative coefficients? The statement 'determines the generating function by cardinality, weight, and the numbers of internal and external edges' is ambiguous without knowing whether the implication is an equality of full generating functions over two alphabets or an equality of evaluations for fixed weights. Please state the weighting hypothesis precisely in the main theorem.
  3. [Two-alphabet independence] A key premise of the claimed result is that the two alphabets in the MacMahon function remain genuinely independent throughout the proof, so that distinct weighted colorings cannot collapse to the same power series. No visible argument establishes this independence. Please provide an explicit, labeled argument showing that the cardinality and weight information can be separated and recovered from the invariant.
minor comments (3)
  1. [Introduction (anticipated)] Please include a definition of the diagonal action and of MacMahon symmetric functions in the introduction, since the abstract assumes familiarity with that setting.
  2. [References] The abstract names Crew's conjecture but does not give a citation to Crew's original paper; please add the reference in the bibliography.
  3. [Notation] Consider defining 'internal edges' and 'external edges' explicitly in the theorem statement; the abstract uses these terms without formal definition.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified from the available abstract; the weighted MacMahon result is presented as a genuine generalization with an unreadable full text.

full rationale

The only text available is the abstract; the full text is supplied as an unreadable corrupted encoding, so no equation or section-level derivation chain can be quoted. The abstract presents a new two-alphabet MacMahon symmetric function for vertex-weighted graphs and claims a theorem that this invariant determines a generating function for vertex subsets by cardinality, weight, and internal/external edge counts. It explicitly frames the result as a generalization of the unweighted Crew conjecture, citing two independent proofs, one of which includes co-author Martin. This is a self-citation, but it concerns the previously established unweighted base case and is not, on the available evidence, load-bearing for the new weighted claim. Nothing in the abstract indicates that the weighted invariant is defined in terms of the target generating function, nor that the theorem merely renames a fitted or assumed quantity. Per the hard rules, circularity cannot be claimed without exhibiting a specific reduction from the paper's own equations, and no such reduction is visible here. The honest finding is therefore no significant circularity, with the caveat that the corrupted full text prevents independent verification of the two-alphabet independence argument.

Assumptions & free parameters 0 free parameters · 3 assumptions · 1 invented entities

All conclusions rest on standard background in symmetric functions and on the previously proved unweighted theorem; the only new object is the two-alphabet invariant itself, whose claimed strength is exactly the theorem under review. No free parameters are fitted, and no physical or geometric entities are postulated.

assumptions (3)
  • standard math Stanley's chromatic symmetric function and the standard theory of symmetric functions are taken as background.
    The paper defines an analogue of the chromatic symmetric function, so the properties of the classical invariant are presupposed as standard background in algebraic combinatorics.
  • domain assumption The unweighted theorem (Crew's conjecture, proved by Aliste-Prieto, Martin, Wagner, Zamora and by Liu, Tang) is correct and serves as the base case or template.
    The abstract states the new result generalizes this theorem; the proof builds on it. Its correctness is assumed without reproving it.
  • standard math MacMahon symmetric functions on two alphabets form a well-defined invariant space for the diagonal action of the symmetric group.
    The new invariant takes values in this space; the multiple-alphabet setup is part of the MacMahon theory the paper adopts.
invented entities (1)
  • The chromatic symmetric MacMahon function, a two-alphabet symmetric function invariant of vertex-weighted graphs independent evidence
    purpose: To record, in a single symmetric function, both the color and cardinality data and the vertex-weight data of a graph, and to recover weighted subset statistics for trees.
    The main theorem provides a checkable handle: for every tree, the invariant is claimed to determine the weighted subset generating function, which can be tested on examples. This is the object of study, not an ad hoc entity introduced to force a conclusion.

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Cite this review

Pith. "Pith review of Chromatic MacMahon symmetric functions of graphs." pith.science (2026). https://pith.science/paper/FG5Y5PFY

@misc{pith2026250800157,
  author       = {Pith},
  title        = {Pith review of: Chromatic MacMahon symmetric functions of graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FG5Y5PFY}},
  note         = {Machine review of arXiv:2508.00157}
}
read the original abstract

A MacMahon symmetric function is an invariant of the diagonal action of the symmetric group on power series in multiple alphabets of variables. We introduce an analogue of the chromatic symmetric function for vertex-weighted graphs, taking values in the MacMahon symmetric functions on two sets of variables, recording information about both cardinalities and weights of vertex sets. We prove that the chromatic symmetric MacMahon function of a tree determines the generating function for its vertex subsets by cardinality, weight, and the numbers of internal and external edges. This result generalizes the one for the unweighted case, first conjectured by Crew and proved independently by Aliste-Prieto--Martin--Wagner--Zamora and Liu--Tang.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 21 canonical work pages

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Reviewed August 6, 2026 · model on record in the stance chip above.