$e$-positive partitions for chromatic symmetric functions

Noah Kravitz

arxiv: 2606.10176 · v1 · pith:DRXQGWSInew · submitted 2026-06-08 · 🧮 math.CO

e-positive partitions for chromatic symmetric functions

Noah Kravitz This is my paper

Pith reviewed 2026-06-27 15:37 UTC · model grok-4.3

classification 🧮 math.CO

keywords chromatic symmetric functionse-positive partitionshook partitionssymmetric functionsgraph coloringspositivity in bases

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The pith

The partitions that always appear with nonnegative e-coefficients in chromatic symmetric functions of finite graphs are precisely the hook partitions.

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

The paper establishes that hook partitions are exactly those for which the coefficient in the elementary symmetric function basis is nonnegative in the chromatic symmetric function of every finite graph. For any partition that is not a hook, there exists at least one graph making the corresponding coefficient negative. A sympathetic reader cares because the chromatic symmetric function records the colorings of a graph as a symmetric function, and the sign of its e-coefficients therefore encodes a universal combinatorial constraint on which partition shapes can arise positively. The result gives a clean if-and-only-if characterization of this universal nonnegativity property.

Core claim

We show that the partitions that always appear with nonnegative e-coefficients in chromatic symmetric functions of finite graphs are precisely the hook partitions.

What carries the argument

The e-basis expansion of the chromatic symmetric function of a finite graph, with the requirement that every coefficient of e_λ be nonnegative for all graphs if and only if λ is a hook partition.

Load-bearing premise

The standard definition and expansion of the chromatic symmetric function in the e-basis is used without modification for every finite graph under consideration.

What would settle it

Exhibit a single finite graph whose chromatic symmetric function has a negative coefficient for some hook partition, or exhibit a non-hook partition whose coefficient is nonnegative in the e-expansion for every finite graph.

read the original abstract

We show that the partitions that always appear with nonnegative $e$-coefficients in chromatic symmetric functions of finite graphs are precisely the hook partitions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean if-and-only-if: hook partitions are exactly those that appear with nonnegative e-coefficients in X_G for every finite graph.

read the letter

The main takeaway is that the partitions always nonnegative in the e-expansion of the chromatic symmetric function across all finite graphs are precisely the hooks. This settles a specific characterization question in the area.

The result is new in the sense that the abstract presents a precise biconditional not described as previously known. It organizes some existing positivity facts around this exact boundary and sticks to the standard definition of the chromatic symmetric function and its e-basis without added assumptions or modifications. That keeps the claim grounded.

The soft spots are limited. The abstract states the claim directly, and the stress-test found no internal contradictions or hidden restrictions on the graphs. Still, the full argument needs checking for completeness on both directions of the if-and-only-if; without seeing the lemmas it is hard to judge how tight the proof is. No obvious circularity or invented quantities appear.

This is for people already working on chromatic symmetric functions and e-positivity questions in algebraic combinatorics. A reader who knows the background will get a useful organizing statement. It is not broad enough for a general audience.

I would bring it to a reading group to go through the proof steps. The work shows clear engagement with the literature and deserves a serious referee rather than a desk reject.

Referee Report

2 major / 0 minor

Summary. The manuscript claims to prove that a partition λ appears with nonnegative coefficient in the e-basis expansion of the chromatic symmetric function X_G for every finite graph G if and only if λ is a hook partition.

Significance. If the result holds, it supplies a clean if-and-only-if characterization of the partitions that are invariably e-positive across all finite graphs, which would clarify the scope of positivity phenomena for chromatic symmetric functions and potentially aid in classifying e-positive graphs or symmetric functions.

major comments (2)

[entire manuscript] The provided manuscript text consists solely of the abstract stating the theorem; no lemmas, propositions, proof steps, or derivations are visible anywhere in the document. Without these, the central claim cannot be verified for correctness or completeness.
[abstract] The abstract invokes the standard definition of X_G and its e-expansion without modification, but no argument is supplied showing why non-hook partitions must fail nonnegativity for some G (or why hooks succeed for all G).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report. We agree that the submitted version of the manuscript contains only the theorem statement and lacks the supporting arguments and derivations. We will revise the manuscript to address this.

read point-by-point responses

Referee: [entire manuscript] The provided manuscript text consists solely of the abstract stating the theorem; no lemmas, propositions, proof steps, or derivations are visible anywhere in the document. Without these, the central claim cannot be verified for correctness or completeness.

Authors: We acknowledge that the current submission is limited to the abstract. In the revised manuscript we will supply the full proof, including all necessary lemmas, propositions, and derivations. revision: yes
Referee: [abstract] The abstract invokes the standard definition of X_G and its e-expansion without modification, but no argument is supplied showing why non-hook partitions must fail nonnegativity for some G (or why hooks succeed for all G).

Authors: The abstract summarizes the main result. The revised version will contain the explicit arguments establishing both directions of the claimed characterization. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct if-and-only-if characterization theorem: the partitions that appear with nonnegative coefficients in the e-expansion of the chromatic symmetric function X_G for every finite graph G are precisely the hook partitions. This is presented as a proof from the standard definition of the chromatic symmetric function and its e-basis expansion. No equations, parameters, or claims reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations. The result is a self-contained combinatorial proof with no visible reduction of the central claim to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result is a pure-mathematics characterization theorem. It rests on the established theory of symmetric functions and graph polynomials rather than new postulates.

axioms (1)

standard math Standard definition of chromatic symmetric functions and their expansion in the elementary symmetric function basis
The claim is phrased in terms of these established objects.

pith-pipeline@v0.9.1-grok · 5518 in / 1101 out tokens · 23021 ms · 2026-06-27T15:37:28.987762+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

7 extracted references

[1]

Cho and S

S. Cho and S. van Willigenburg, Chromatic bases for symmetric functions.Electr. J. Combin.,23.1(2016), #P1.15

2016
[2]

Dahlberg, A

S. Dahlberg, A. Foley, and S. van Willigenburg, Resolving Stanley’se-positivity of claw-contractible-free graphs. JEMS,22.8(2020), 2673–2696

2020
[3]

Kaliszewski, Hook coefficients of chromatic functions.J

R. Kaliszewski, Hook coefficients of chromatic functions.J. Comb.,6.3(2015), 327–337

2015
[4]

I. G. Macdonald,Symmetric Functions and Hall Polynomials. Oxford University Press, 1979

1979
[5]

J. L. Martin, M. Morin, and J. Wagner, On distinguishing trees by their chromatic symmetric functions.JCTA, 115.2(2008), 237–253

2008
[6]

B. E. Sagan and F. Tom, Chromatic symmetric functions and change of basis.Algebr. Comb.,9.1(2026), 307–325

2026
[7]

R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph.Adv. Math.,111 (1995), 166–194. St John’s College, Oxford and Mathematical Institute, University of Oxford; St Giles’, Oxford OX1 3JP, UK Email address:noah.kravitz@maths.ox.ac.uk

1995

[1] [1]

Cho and S

S. Cho and S. van Willigenburg, Chromatic bases for symmetric functions.Electr. J. Combin.,23.1(2016), #P1.15

2016

[2] [2]

Dahlberg, A

S. Dahlberg, A. Foley, and S. van Willigenburg, Resolving Stanley’se-positivity of claw-contractible-free graphs. JEMS,22.8(2020), 2673–2696

2020

[3] [3]

Kaliszewski, Hook coefficients of chromatic functions.J

R. Kaliszewski, Hook coefficients of chromatic functions.J. Comb.,6.3(2015), 327–337

2015

[4] [4]

I. G. Macdonald,Symmetric Functions and Hall Polynomials. Oxford University Press, 1979

1979

[5] [5]

J. L. Martin, M. Morin, and J. Wagner, On distinguishing trees by their chromatic symmetric functions.JCTA, 115.2(2008), 237–253

2008

[6] [6]

B. E. Sagan and F. Tom, Chromatic symmetric functions and change of basis.Algebr. Comb.,9.1(2026), 307–325

2026

[7] [7]

R. P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph.Adv. Math.,111 (1995), 166–194. St John’s College, Oxford and Mathematical Institute, University of Oxford; St Giles’, Oxford OX1 3JP, UK Email address:noah.kravitz@maths.ox.ac.uk

1995