When is the chromatic quasisymmetric function symmetric?

Joseph Pappe; Kyle Salois; Maria Gillespie

When is the chromatic quasisymmetric function symmetric?

T0 review · 0 major / 7 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that the chromatic quasisymmetric function of a connected acyclic directed graph is symmetric only when the graph has exactly one source and one sink, that directed paths are the only trees with symmetric CQF, and that…

desk verdict The product theorem and the DAG nonsymmetry result are real and appear sound; the internal-review concern about Lemma 4.10 does not hold up on reading. read the letter →

arxiv 2412.10556 v2 pith:IROJA52D submitted 2024-12-13 math.CO

Maria Gillespie , Joseph Pappe , Kyle Salois This is my paper · ORCID

classification math.CO MSC 05E0505C1505C20

keywords chromaticquasisymmetricfunctionfunctionssymmetricdirectedacyclicgraphsgraphcoloringschaindecompositionsmountaine-positivity

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 asks which graphs have a chromatic quasisymmetric function (the $q$-weighted generating function of proper colorings) that is actually symmetric rather than merely quasisymmetric. It proves three things: a product of two quasisymmetric functions in infinitely many variables is symmetric only if both factors are, so the symmetry question splits into connected components; every connected directed acyclic graph with more than one source or more than one sink has a nonsymmetric CQF, which makes directed paths the only trees with symmetric CQF; and a new family called mixed mountain graphs always has symmetric CQF. The first result matters because it removes disjoint unions from consideration, the second settles the tree question, and the third provides a nontrivial family with which to test whether every symmetric CQF is e-positive.

What carries the argument

The main objects are the chromatic quasisymmetric function $X_G(x;q)$, which records proper colorings by the monomial $x_{\kappa(1)}\cdots x_{\kappa(n)}$ times $q^{\#\text{ascents}}$, and the monomial quasisymmetric functions $M_\alpha$ in which its non-symmetry is detected. For the negative results, the load-bearing mechanism is a comparison of the coefficients of two $M_\alpha$'s at the maximal $q$-power $q^{|E|}$: a chain decomposition of the graph (minimum number of disjoint chains equal to the largest antichain) is used to build a map between the two weight classes whose failure to be surjective is visible in these coefficients. For the positive results, the machinery is a family of ascent-preserving bijections—an involution on colorings whose $(a,a+1)$-colored subgraph avoids the bottom edge, and 'cycle', 'reflect', and 'swap' maps on colorings that include it—which together swap the counts of any two adjacent colors.

What would settle it

Enumerate all labeled connected directed acyclic graphs with exactly two sources and two sinks up to eight or nine vertices and, for each, compute the coefficients of $M_{(1^k,a,1^{n-k-a})}$ and $M_{(a,1^{n-a})}$ at $q^{|E|}$ in $X_G(x;q)$; the theorem predicts the two coefficients always differ, so a graph where they agree would refute the classification.

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Extended reading notes

Core claim

The central claim is a structural dichotomy. On the nonsymmetric side, for any connected directed acyclic graph whose sources outnumber one or whose sinks outnumber one, the coefficient of $q^{|E|}$ in the expansion of $X_G(x;q)$ in the monomial quasisymmetric basis distinguishes two weight classes: the maximum-ascent colorings of weight $(1^k,a,1^{n-k-a})$ force the weight-$(a,1^{n-a})$ class to miss $q^{|E|}$, so the function cannot be symmetric. On the symmetric side, the paper shows that every mixed mountain graph—a cycle of cliques and bottomless cliques strung together and oriented left-to-right—admits ascent-preserving bijections that swap the counts of any two adjacent colors, which is exactly what symmetry of a CQF requires. Together with the product theorem, these results reduce the symmetry problem to connected graphs and give the first tree classification.

Load-bearing premise

The proof that the map between the two weight classes is injective but not surjective rests on the assumption that the recolored vertex of color $k+1$ is the source of every chain that changes; the argument only shows each changed chain contains a color-$k+1$ vertex, and if a chain also contains one of the singleton colors $2,\dots,k$, its source has the smaller color and the inversion step breaks.

Editorial extensions

If this is right

Symmetry of a disjoint union forces each connected component's CQF to be symmetric, so the classification needs only connected graphs.
A connected directed acyclic graph with two or more sources or two or more sinks has a nonsymmetric CQF; a symmetric CQF forces exactly one source and one sink and a directed path through every vertex.
Among oriented trees, the directed path is the unique graph with a symmetric CQF.
Among directed acyclic cycles, exactly the naturally oriented cycles are symmetric.
The mixed mountain graphs are a new infinite family of symmetric CQFs that are not generally natural unit interval graphs.

Reading between the lines

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

If the dichotomy in the nonsymmetry theorem holds, the remaining open classification is the connected single-source single-sink directed acyclic graphs that contain a directed Hamiltonian path; mixed mountain graphs are one family inside that class, and unit interval orders appear to be another up to eight vertices.
The paper's open question—whether every symmetric CQF is e-positive—can be tested on mixed mountain graphs: if any mixed mountain graph fails e-positivity the answer is no, and if all pass that supports a positive answer.
The product theorem genuinely depends on having infinitely many variables: in finitely many variables, $x_1^2x_2\cdot x_1x_2^2 = x_1^3x_2^3$ is symmetric while the factors are not, so any finite-variable analogue would need a different mechanism.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I read the paper with the reader's Lemma 4.10 worry in mind. I think it is a misreading. The inversion does not require knowing where the k+1 sat before sorting; the map acts on each chain by replacing the unique k+1 with 1 and then sorting. Since every chain in a maximum-ascent coloring is strictly increasing, the output chain is the sorted version of that multiset. Replacing the unique 1 with k+1 and sorting recovers the original chain, whatever its length or the position of k+1. The non-surjectivity argument also stands: phi never introduces a new k+1, so a preimage of a coloring where a vertex of S(G) is colored k+1 would need that vertex to have color k+1 already, which Lemma 4.8 forbids. So the main DAG theorem is supported.

What is genuinely new: Theorem 1.4 is a striking factorization result in infinite variables, and it cleanly reduces the symmetry question to connected graphs. Theorem 1.6 and its tree corollary answer an open question and extend the earlier path/star results by a real margin. The mixed mountain family is a nice addition, especially because it is not unit interval in general, so it broadens the known symmetric territory. The proofs are long but mostly careful; the mountain section has a lot of moving parts and would benefit from more figures or a few more checks, but I do not see a gap.

Soft spots in proportion: Lemma 4.5's proof that S(G) is nonempty is terse enough to need a sentence or two extra; the connectivity argument is not hard but as written it can trip a reader. The Sage computation is mentioned without code or data; that is a reproducibility nit, not a mathematical flaw. The mountain graph swap map relies on a case analysis that takes some effort to verify, but it appears correct.

Who should read it: anyone working on chromatic symmetric functions or on quasisymmetric factorization more broadly. It is a solid contribution that deserves a serious referee. I would send it out; if I were the referee, I would accept after minor revision, mostly for exposition and for supplying the computational data.

Referee Report

0 major / 7 minor

Summary. The paper studies when the chromatic quasisymmetric function X_G(x;q) of a labeled graph is symmetric. It proves a product theorem (Theorem 1.4): if a product of two quasisymmetric functions in countably infinitely many variables is symmetric, then both factors are symmetric, which reduces the symmetry question for CQFs to connected graphs. It then proves (Theorem 1.6) that any connected directed acyclic graph with more than one source or sink has a nonsymmetric CQF, yielding as a corollary that a tree has a symmetric CQF if and only if it is a directed path. Finally, it introduces a family of mixed mountain graphs and proves their CQFs are symmetric (Theorem 1.8) by constructing ascent-preserving involutions and automorphisms that swap adjacent color multiplicities. The authors also report Sage verification on all connected labeled graphs up to eight vertices.

Significance. If the results stand, Theorem 1.6 settles the tree question posed in [4] and, together with Theorem 1.4, reduces the global symmetry question to connected DAGs with exactly one source and one sink. Theorem 1.4 is a clean standalone statement with useful consequences beyond the graph application. The proofs are mostly explicit and combinatorial, with no fitted parameters, no circular dependencies, and a constructive treatment of the mixed mountain graph family. The main limitation is that Theorem 1.8 establishes symmetry but not e-positivity, so the paper does not directly resolve Question 1.3; the computational evidence is suggestive but not load-bearing for the theorems.

minor comments (7)

[§3, Theorem 1.4 proof] The claim that QSym is a UFD is justified by the text 'there are a finite number of generators of each degree, so any given polynomial lies in a finitely generated ring that has unique factorization'; finite generation alone does not imply unique factorization. Please cite the algebraic independence of the Hazewinkel generators, or Grinberg–Reiner Corollary 6.5.33, explicitly, since this step is load-bearing for Theorem 1.4.
[§3, Theorem 1.4, Case 2] The induction hypothesis is stated for a product of two factors, but the proof concludes that every irreducible factor a_i is symmetric from u = a_1...a_r. This requires iterating the two-factor statement degree by degree; the iteration is straightforward but should be said explicitly.
[§4.2, Lemma 4.5] The proof that S(G) is nonempty is compressed: it says that if every sink were paired with a unique source then G would be disconnected, without explaining why two source-to-sink paths in a connected graph must merge before reaching distinct sinks. Spelling out the first merge point would make the argument complete.
[§4.2, Lemma 4.10] The injectivity argument is very terse. It would help to state explicitly that unchanged chains still contain color k+1, changed chains have lost it, and the inverse simply recolors the source of a changed chain with k+1 and re-sorts; this makes the inversion valid regardless of where k+1 sat in the original chain.
[§5.3, Proposition 5.26] The number of swap applications should be m(p-m), the number of adjacent pairs needed to reverse the order of the m k-cliques and p-m bottomless k+1-cliques; the printed expression 'a(p-a)' appears to use the color parameter a and should be corrected.
[§5.2–5.3] The terminology 'bottomless k+1-mountain' used in the introduction and Theorem 1.8 is not obviously the same as the graph B_{p,k} defined in §5.2, where a k-clique has the edge between its lower vertices removed. Please reconcile the definitions so the number of vertices in a bottomless mountain is unambiguous.
[Miscellaneous] There are several small typos, including 'Supopse' in Remark 3.9 and 'CHROMA TIC' in the running header, which should be corrected in a final revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained and rests on external standard theorems and explicit constructive maps.

full rationale

The paper's central claims are not circular. Symmetry of a chromatic quasisymmetric function is tested by comparing coefficients in the monomial quasisymmetric basis, and the proofs proceed by explicit ascent-preserving constructions (the maps phi, cycle, reflect, and swap) plus standard external ingredients: Dilworth's theorem, Hazewinkel's generator theorem for QSym, Lam-Pylyavskyy's UFD/factorization theorem, and Shareshian-Wachs results on unit interval graphs. No parameter is fitted to data and then renamed a prediction; no theorem is assumed in the form of its own conclusion; and no load-bearing step is justified by self-citation of the present authors. Theorem 1.4 is proved from Hazewinkel's generators and the UFD property of QSym, not from the desired statement. Theorem 1.6 is a coefficient-comparison argument: Lemma 4.2 compares M_(a,1,b) and M_(b,1,a), Lemma 4.4 handles large antichains, and Lemma 4.10 constructs an injective non-surjective map between color classes. Even if the reader's concern about Lemma 4.10's inversion were a real gap, that would be a correctness issue, not circularity, because the map does not build the target nonsymmetry into its definition. The mixed-mountain symmetry results use ascent-preserving bijections and the external unit-interval-graph machinery of Shareshian-Wachs; they do not presuppose that the graphs in question have symmetric CQFs. The paper is therefore self-contained against external benchmarks, and no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

All axioms are standard cited results; no ad hoc assumptions, fitted constants, or newly invented objects are introduced. The only structurally novel premise is the fixed Dilworth chain decomposition, which is a standard tool.

assumptions (5)

standard math Hazewinkel's theorem: QSym_Z is a polynomial algebra generated by lambda_n(M_alpha) for Lyndon words.
Invoked in Section 3 to establish unique factorization and a finite-generation argument in the proof of Theorem 1.4.
standard math Lam-Pylyavskyy Theorem 8.1: irreducible factors in K of a quasisymmetric function are quasisymmetric.
Used in Corollary 3.8 to extend the product theorem to arbitrary power series in K.
standard math Dilworth's theorem: minimum chain decomposition equals maximum antichain size.
Used in Section 4 to fix a chain decomposition R with one source and sink per chain for the map phi.
domain assumption Shareshian-Wachs theorem: natural unit interval graphs have symmetric CQF.
Used for directed paths in Corollary 4.12 and as a base for the involution in Proposition 5.3.
domain assumption Ellzey-Wachs theorem: naturally oriented cycles have symmetric CQF.
Used in Corollary 4.13 for the characterization of directed acyclic cycles.

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Pith. "Pith review of When is the chromatic quasisymmetric function symmetric?." pith.science (2026). https://pith.science/paper/IROJA52D

@misc{pith2026241210556,
  author       = {Pith},
  title        = {Pith review of: When is the chromatic quasisymmetric function symmetric?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IROJA52D}},
  note         = {Machine review of arXiv:2412.10556}
}

abstract

We investigate the problem of when a chromatic quasisymmetric function (CQF) $X_G(x;q)$ of a graph $G$ is in fact symmetric. We first prove the remarkable fact that if a product of two quasisymmetric functions $f$ and $g$ in countably infinitely many variables is symmetric, then in fact $f$ and $g$ must be symmetric. This allows the problem to be reduced to the case of connected graphs. We then show that any labeled graph having more than one source or sink has a nonsymmetric CQF. As a corollary, we find that all trees other than a directed path have a nonsymmetric CQF. We also show that a family of graphs we call ''mixed mountain graphs'' always have symmetric CQF.

Figures

Figures reproduced from arXiv: 2412.10556 by the authors.

**Figure 1.** The path graph of length 1 with its labels is shown at left above; two colorings are shown at middle and right, with the middle having one ascent and the right hand diagram having none. Date: September 2, 2025. All authors were partially supported by NSF DMS award number 2054391. 1 arXiv:2412.10556v2 [math.CO] 1 Aug 2025 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗

**Figure 2.** A graph G shown at left, for which XG(x; q) is not symmetric. Three of its colorings with maximal number of ascents are shown to the right. The Stanley–Stembridge conjecture [31, 33] states that certain chromatic symmetric functions are e-positive, meaning that they expand positively in the basis of elementary symmetric functions (see Section 2.1). In particular, the subset of graphs in this conjecture are the incom… view at source ↗

**Figure 3.** A mixed mountain graph for k = 4. Note that this does not hold for finitely many variables; we have that x 2 1x2 · x1x 2 2 = x 3 1x 3 2 is symmetric in two variables. Since CQF’s are multiplicative across disjoint union of graphs, we immediately have the following. Corollary 1.5. Suppose G is a graph with two or more connected components, and XG(x; q) is symmetric. Then the CQF of each connected component is symmetr… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: The (5, 4)-mountain graph Notably, these graphs are not natural unit interval graphs unless p = 2 and k = 2 (or a relabeling of a natural unit interval graph when p = 2 and k ≥ 3). We will show that if G is a (p, k)-mountain graph, then XG(x; q) is symmetric. In the ca…

**Figure 5.** Figure 5: Above, an element κ ∈ L2,3(G), and below, the output cycle(κ) ∈ L1,2(G). Next, suppose that C is a clique of κ containing at least one (and thus, exactly one) vertex colored 1. We have two cases. If v is a lower vertex, then v cannot be the leftmost vertex v1 or rightm…

**Figure 6.** Figure 6: The output reflect(cycle(κ)) for κ as in [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]

**Figure 7.** Figure 7: The (5, 4)-bottomless mountain graph We keep the same ordering on the vertices of Bp,k as for Mp,k as well as the definitions of the bottom edge, lower vertices, and upper vertices. To show the chromatic quasisymmetric function [PITH_FULL_IMAGE:figures/full_fig_p015_7.png]

**Figure 8.** Figure 8: An example of the swap map, showing that we can swap a k-clique with an adjacent bottomless k+ 1-clique without changing the chromatic quasisymmetric function. of Bp,k is symmetric, it suffices to show that the ascent-preserving automorphisms presented in Section 5.1 f…

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Chromatic quasisymmetric functions for signed graphs
math.CO 2025-08 conditional novelty 6.0 of 10

A new chromatic quasisymmetric invariant for directed signed graphs and an algebra SQSym of signed quasisymmetric functions are defined and studied.

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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Chromatic Quasisymmetric Functions of the Path Graph

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work page doi:10.1007/s00026-025-00762-1 2025

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Chromatic symmetric functions from the modular law

Alex Abreu and Antonio Nigro. “Chromatic symmetric functions from the modular law”. In: J. Comb. Theory Ser. A180 (2021), Paper No. 105407, 30. issn: 0097-3165,1096-0899. doi: 10.1016/j.jcta.2021.105407

arXiv 2021

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LLT polynomials, chromatic quasisymmetric func- tions and graphs with cycles

Per Alexandersson and Greta Panova. “LLT polynomials, chromatic quasisymmetric func- tions and graphs with cycles”. In: Discrete Math. 341.12 (2018), pp. 3453–3482. issn: 0012- 365X,1872-681X. doi: 10.1016/j.disc.2018.09.001

doi:10.1016/j.disc.2018.09.001 2018

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The chromatic symmetric function of a graph centred at a vertex

Farid Aliniaeifard, Victor Wang, and Stephanie van Willigenburg. “The chromatic symmetric function of a graph centred at a vertex”. In: Electron. J. Combin.31.4 (2024), Paper No. 4.22,

work page 2024

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Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties

Patrick Brosnan and Timothy Y. Chow. “Unit interval orders and the dot action on the cohomology of regular semisimple Hessenberg varieties”. In: Adv. Math.329 (2018), pp. 955–

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Positivity of chromatic symmetric functions associated with Hessenberg functions of bounce number 3

Soojin Cho and Jaehyun Hong. “Positivity of chromatic symmetric functions associated with Hessenberg functions of bounce number 3”. In: Electr. J. Comb.29.2 (2022), Paper No. 2.19,

work page 2022

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On e-positivity and e-unimodality of chromatic quasi-symmetric functions

Soojin Cho and Jisun Huh. “On e-positivity and e-unimodality of chromatic quasi-symmetric functions”. In: SIAM J. Discrete Math.33.4 (2019), pp. 2286–2315.issn: 0895-4801,1095-7146. doi: 10.1137/18M1216201

work page doi:10.1137/18m1216201 2019

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Chromatic symmetric func- tions of Dyck paths and q-rook theory

Laura Colmenarejo, Alejandro H. Morales, and Greta Panova. “Chromatic symmetric func- tions of Dyck paths and q-rook theory”. In: European J. Comb.107 (2023), Paper No. 103595,

work page 2023