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Chromatic quasisymmetric functions for signed graphs

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Pith's one-line read Signed graphs get a chromatic quasisymmetric invariant that expands over signed permutations in the fundamental family of SQSym.

desk verdict New signed quasisymmetric invariant and algebra are solid, but the key expansion theorem has a gap at the middle descent. read the letter →

arxiv 2508.20200 v1 pith:OJZC2RI7 submitted 2025-08-27 math.CO

Jean-Christophe Aval , Raquel Melgar This is my paper · ORCID
classification math.CO MSC 05E0505C2205C15
keywords signedgraphschromaticquasisymmetricfunctionshyperplanearrangementsacyclicorientations(Pω)-partitionspermutationsfundamentalbasis
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 sets out to extend the classical chromatic symmetric function and its directed-graph refinement to signed graphs, where edges carry a + or − sign and colorings assign integers with the condition that adjacent vertices cannot have equal or opposite colors according to the edge sign. Its new object is $X_{\vec{\Sigma}}(x;t)$, a polynomial in $t$ whose coefficients record, for each proper signed coloring, the number of edges where the coloring disagrees with the chosen orientation; at $t = 1$ it recovers the known signed chromatic symmetric function. The authors show that this invariant lives in $\mathrm{SQSym}$, the algebra of signed quasisymmetric functions, which they construct with a monomial basis indexed by bicompositions and a fundamental family built from signed $(P,\omega)$-partitions. The main result is an expansion of $X_{\vec{\Sigma}}(x;t)$ over signed permutations in that fundamental family, with the $t$-exponent given by a signing-weighted inversion statistic and the descent set coming from ranks in the associated signed poset. This gives a signed analogue of the directed chromatic quasisymmetric theory and connects signed graph coloring to hyperplane arrangements through the chamber–orientation bijection.

What carries the argument

The load-bearing object is the invariant $X_{\vec{\Sigma}}(x;t)$ itself, defined by summing $t^{\mathrm{asc}(\kappa)} x^\kappa$ over proper signed colorings, where $\mathrm{asc}(\kappa)$ counts edges $e=\{u,v\}$ on which $\tau(u,e)\kappa(u)+\tau(v,e)\kappa(v)>0$. It is a signed quasisymmetric function, an element of the algebra $\mathrm{SQSym}$ of bounded-degree series in variables $(\ldots, x_{-1}, x_0, x_1, \ldots)$ that are invariant under the quasisymmetrizing action of the symmetric group. The expansion is carried by the fundamental family $F^\varepsilon_S$, the $(P,\omega)$-partition enumerators of signed labeled chains: each $F^\varepsilon_S$ is a sum over weakly increasing index sequences with strict steps prescribed by $S$ and sign pattern $\varepsilon$. The proof machinery is the hyperplane arrangement $H_\Sigma$ who

What would settle it

For a small signed graph, compute both sides of Theorem 4.2: the chamber sum (2.5) and the signed-permutation sum. If they differ for any graph, for example the two-vertex graph with one positive and one negative edge, the expansion fails. A direct count of $|SS_d|$ against the number of pairs (acyclic orientation $\tau'$, linear extension of $P_{\tau'}$) used in the proof would also expose the assumed bijection.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that every directed signed graph $\vec{\Sigma} = (\Sigma, \tau)$ on $d$ vertices has a canonical $t$-refinement $X_{\vec{\Sigma}}(x;t) = \sum_{\kappa \text{ proper}} t^{\mathrm{asc}(\kappa)} x^\kappa$ of the signed chromatic symmetric function, and that this refinement expands as $X_{\vec{\Sigma}}(x;t) = \sum_{\pi \in SS_d} t^{\mathrm{inv}_{\vec{\Sigma}}(\pi)} F^{\mathrm{sgn}(\pi)}_{\mathrm{DES}_\Sigma(\pi)}$ in the fundamental family of $\mathrm{SQSym}$, where $\mathrm{inv}$ counts directed inversions of $\pi$ in the symmetric double cover of $\Sigma$ and $\mathrm{DES}_\Sigma(\pi)$ is a descent set defined by ranks in the associated signed poset. The proof interprets the invariant chamber-by-chamber through the hyperplane arrangement $H_\Sigma$, expresses each chamber's contribution as a signed $(P,\omega)$-partiti

Load-bearing premise

The proof of the main expansion moves from a sum over acyclic orientations to a sum over all signed permutations by relying on an unproved bijection between signed permutations and linear extensions of acyclic orientations of the signed graph; if that bijection is wrong, the formula double-counts or misses terms.

Editorial extensions

If this is right

  • Specializing t=1 turns X_{\vec\Sigma}(x;t) into the known signed chromatic symmetric function, so the new invariant is a genuine refinement of signed graph coloring.
  • The fundamental-family expansion gives a positive formula for X_{\vec\Sigma} in terms of signed permutations, making the invariant algorithmically computable from orientation data alone.
  • SQSym is a graded algebra with monomial basis indexed by bicompositions, and its Hilbert series (1−t)/(1−4t+2t^2) gives dimensions 1, 3, 10, 34, ...; the product of fundamental elements is governed by a shuffle rule.
  • For acyclic orientations containing at least one negative edge, X_{\vec\Sigma}(x;t) is never signed-symmetric, so full symmetry is restricted to special families such as the switched circular indifference digraphs studied in the paper.

Reading between the lines

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

  • Editorial: the proof of the main expansion assumes a bijection between signed permutations and linear extensions of acyclic orientations of the signed graph; the paper does not prove this set-theoretic step, so the expansion's correctness rests on that missing verification.
  • If the fundamental expansion survives scrutiny, SQSym is likely to fit into a type-B or free-quasisymmetric-function framework, giving a signed analogue of the quasi-shuffle Hopf algebra that the paper only sketches.
  • A natural testable extension is to ask whether X_{\vec\Sigma}(x;t) distinguishes signed graphs that the t=1 specialization cannot, mirroring the open distinguishing-power questions for the unsigned invariant.
  • Setting x_0=0 recovers the zero-free signed coloring invariant; the expansion then suggests a zero-free analogue of the (Σ,π)-descent statistics, which could be studied combinatorially on its own.
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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

2 major / 6 minor

Summary. The paper introduces a chromatic quasisymmetric invariant X_{−→Σ}(x;t) for directed signed graphs, refining the signed chromatic symmetric function of Zaslavsky and giving a signed analogue of Ellzey's directed chromatic quasisymmetric function. The invariant is defined via proper signed colorings weighted by ascents, with a hyperplane-arrangement interpretation. The authors then define and study the algebra SQSym of signed quasisymmetric functions, construct a monomial basis, introduce a fundamental family F indexed by signed chains, and prove a product formula in that family. The central deliverable is Theorem 4.2, which expands X_{−→Σ}(x;t) in the fundamental family with coefficients t^{inv_{−→Σ}(π)} and descent sets DES_Σ(π). Section 5 studies when the invariant lies in SSym[t], proving symmetry for a switched family of circular indifference signed graphs and giving counterexamples to broader symmetry.

Significance. If the main theorem is correct, this is a substantial contribution to the algebraig combinatorics of signed graphs. The invariant naturally specializes to the known signed chromatic symmetric function at t=1 and to the zero-free variant by adding negative loops. The paper provides a new algebra SQSym with a monomial basis, a dimension formula and generating function (3.5), and a fundamental family that interacts well with the expansion of the new invariant. The hyperplane-arrangement viewpoint is well motivated, and the definitions are for the most part clear and self-consistent. The paper gives concrete examples and tables that make the constructions accessible. The main concern is a gap in the proof of the central expansion, which is load-bearing for the paper's main claim.

major comments (2)
  1. [Definition 4.1 and Theorem 4.2] The Σ-descent at i=0 is not defined: rank(Σ,π)(m) is defined only for m∈{±1,...,±d}, but the i=0 case compares rank(Σ,π)(π_0)=rank(Σ,π)(0). In the proof of Theorem 4.2, the equality DES(ωτπ)=DESΣ(π) is argued only for m=1,...,d-1. For i=0, the middle strict edge of F^{sgn(π)}_{DESΣ(π)} is governed by whether ω(π(1))<0 (see Eq. (3.9)), which depends on the position of π(1) in the order-reversing labeling, not on a rank of 0. The final identification DES(ωτπ)=DESΣ(π) is therefore unverified exactly where F has its critical middle edge. This gap affects the central expansion and needs to be repaired by a correct definition of the 0-descent or by a direct proof of the m=0 equivalence.
  2. [Theorem 5.4] The proof of the symmetry theorem is incomplete. The involution Φ_i is described in three cases, but the rotation argument in the even-length path containing v' is not precisely specified, and the statement 'It is to check that Φ_i is an involution and that the number of ascents is preserved' is not a proof. Moreover, the generation statement 'SS is generated by π_i := (i, i+1)(i, i+1)' is evidently incorrect as written (that product is the identity), and π0 is later written as (1,-1). Since Theorem 5.4 is a stated result of the paper, these details need to be completed and corrected.
minor comments (6)
  1. [Proposition 3.7] The product formula for monomial signed quasisymmetric functions is stated with 'The proof is omitted'. Since this is a structural property of SQSym, either a proof or a detailed reference to the cited quasi-shuffle argument should be included.
  2. [Eq. (3.9)] The condition 'ij < ij+1 if i∈S' should presumably be 'if j∈S'. Please clarify the indexing of strict edges in the formula for F^ϵ_S.
  3. [Theorem 4.2 proof] The equality asc_{−→Σ}(τπ)=inv_{−→Σ}(π) is asserted as 'not difficult to see' without proof. A short argument should be supplied, since it is part of the main theorem's statement.
  4. [Theorem 4.2 proof] The asserted 'bijection between SS_d and the linear extensions of acyclic orientations of Σ' is stated without justification. It is true (via the disjoint union over τ' of L(P_{τ'},e)), but a sentence explaining the disjoint-union argument would remove ambiguity.
  5. [Table 1] The table in Example 4.3 is hard to read: the same strings '2112' and '1221' appear repeatedly, but the statistics and sign vectors differ. The signed permutations should be displayed unambiguously (e.g., with explicit overlines or sign vectors) to avoid confusion.
  6. [Throughout] There are several small typos, e.g., the definition of π_i in Section 5 and SQQym/SQqym in Corollary 3.5. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the invariant, algebra, and expansion are derived from independent definitions and external theorems.

full rationale

The paper's central invariant X_{→Σ}(x;t) is defined independently in Definition 2.14 as a weighted sum over proper colorings. The algebra SQSym is constructed separately via a quasisymmetrizing action (Definition 3.1) and the fundamental family F is defined as (P,ω)-partition enumerators of signed chains (Definition 3.18), without reference to X. Theorem 4.2 is proved through Zaslavsky's chamber/acyclic-orientation bijection (Theorem 2.12), Stanley's decomposition of (P,ω)-partitions into chains (Proposition 3.14), and a reindexing of acyclic orientations by signed permutations. No parameter is fitted to data and then called a prediction; the descent set DES_Σ and inversion statistic inv are defined directly from the graph and permutation. The only self-citations ([ADM22] for an analogous unsigned construction, [ABB04] for a remark) are not load-bearing. The proof has expositional gaps—the 'bijection' between SS_d and linear extensions of acyclic orientations is really a disjoint-union partition, and the i=0 case of DES_Σ is not handled by Definition 4.1—but these are correctness concerns, not circular reductions. The claimed expansion is not equivalent to its inputs by construction.

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

No free parameters are fitted; the paper introduces a new invariant and a new algebra, relying on standard theorems from Stanley, Zaslavsky, Hivert, and Ellzey. No physical entities are postulated.

assumptions (3)
  • domain assumption Zaslavsky's theorem: chambers of HΣ are in bijection with acyclic orientations of Σ (Theorem 2.12, cited as [Z91]).
    This is the foundation for the geometric interpretation (2.5) and for the use of chambers in the expansion.
  • standard math Stanley's theory of (P,ω)-partitions: the decomposition of (P,ω)-partitions into linear extensions (Proposition 3.14, cited to [S72]).
    Used to obtain the fundamental expansion of Γ_P,ω in terms of F.
  • standard math Hivert's characterization of quasisymmetric functions under the quasisymmetrizing action (Section 3.1, [Hi00]).
    Used to define SQSym as invariants under the action and to prove it is an algebra.

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Pith. "Pith review of Chromatic quasisymmetric functions for signed graphs." pith.science (2026). https://pith.science/paper/OJZC2RI7

@misc{pith2026250820200,
  author       = {Pith},
  title        = {Pith review of: Chromatic quasisymmetric functions for signed graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OJZC2RI7}},
  note         = {Machine review of arXiv:2508.20200}
}
abstract

In 1995, Stanley introduced the chromatic symmetric function of a graph, which specializes to its chromatic polynomial, and which has been the focus of intense research. In 2017, Shareshian, Wachs, and Ellzey defined a refinement of this function for a directed graph, that appears to be in $QSym$, the algebra of quasisymmetric functions, which is of great interest in algebraic combinatorics. Our goal is to extend this work to signed graphs, taking into account the perspective of the hyperplane arrangement associated with a signed graph, developed by Zaslavsky. We introduce the signed chromatic quasisymmetric invariant, and obtain structural properties. As a consequence, we define and study $SQSym$, the algebra of signed quasisymmetric functions.

Figures

Figures reproduced from arXiv: 2508.20200 by the authors.

Figure 1
Figure 1. A signed graph, where the negative edges have been drawn with dashed lines, with one of its frame circuits in gray. A switching function for Σ = (V, E, σ) is a map ν : V → {+, −}. Such a function acts on signatures and thus on signed graphs as follows. A signature σ may be switched by ν to get σ ν : E → {+, −} defined as σ(e) := ν(u)σ(e)ν(v) where u, v ∈ V are the endpoints of e ∈ E. And Σν := (V, E, σν ) may be cal… view at source ↗
Figure 2
Figure 2. A signed graph Σ and Σν where ν : V → {+, −} sends every vertex to + except v1 and v4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Possible orientations of an edge an endpoint of e ∈ E. The interpretation of τ is the following: if τ (u, e) = + then the indicence (u, e) points into the vertex u and if τ (u, e) = −, the incidence points away from u. Given a signed graph Σ = (G, σ) we call an orientation of Σ a bidirection τ of G = (V, E) verifying σ(e) = −τ (u, e)τ (v, e) (2.1) on every edge e = {u, v} ∈ E. A directed signed graph −→Σ is a pair (… view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: A signed graph and one of its possible orientations. Definition 2.2. Given a signed graph Σ = (V, E, σ), a coloring of Σ is any map κ : V → Z. We say that κ is a proper coloring if κ(u) ̸= σ(e)κ(v) for all e ∈ E (2.2) where u, v ∈ V are the endpoints of e. Given a swit…
Figure 5
Figure 5. Figure 5: A signed graph and its corresponding symmetric graph Definition 2.8. A proper coloring of a symmetric graph G = (V, E) with V = {u1, u−1, . . . , ud, u−d} is a map κ : V → Z verifying • κ(ui) = −κ(−ui) for every i ∈ {1, . . . , d} • κ(ui) ̸= κ(uj ) for every edge {ui ,…
Figure 6
Figure 6. Figure 6: Hyperplane arrangement and acyclic orientations of Exam￾ple 2.13 where asc(κ) := #{e = {vi , vj} ∈ E : τ (vi , e)κ(vi) + τ (vj , e)κ(vj ) > 0}. As required, this definition is compatible with the interpretation in terms of hyperplane arrangements, as: X−→Σ (x;t) = X C …
Figure 7
Figure 7. Figure 7: A directed signed graph and its corresponding arrangement in R 2 . We start with an easy statement. Proposition 2.16. If −→Σ 1 + −→Σ 2 denotes the disjoint union of two directed signed graph −→Σ 1 and −→Σ 2, then one has: X−→Σ1+ −→Σ2 (x;t) = X−→Σ1 (x, t)X−→Σ2 (x, t) […
Figure 8
Figure 8. Figure 8: The Hasse diagram of a signed poset and three of its linear extensions, that we will note as 1342, 4123 and 4213 respectively. When drawing a chain, it is sufficient to specify half of it. Definition 3.10. A permutation π of Z is called a signed permutation if π(−i) = …
Figure 9
Figure 9. Figure 9: The inequalities induced by the ω on the left are illustrated in the diagram on the right. As a direct consequence from Stanley’s result [S72, Lemma 6.1] on usual (P, ω)- partitions we have the following result. Proposition 3.14. Let L(P, ω) be the set of linear extens…
Figure 10
Figure 10. Figure 10: + − − + − + + − + = + + + + + + + + + + − − + − + + − + + + + + + + + Example 3.22. Let d = 4 then F (+++−) {1,3} , F (−++−) {1,2} , F (−−++) {0,2} and F (−+−+) {0,3} are respectively the (P, ω)-partition enumerators of the chains in [PITH_FULL_IMAGE:figures/full_fig…
Figure 11
Figure 11. Figure 11: Thus F ϵ S has the following form F ϵ S = X 0≤i1≤···≤id ij<ij+1 if i∈S 0<i1 if 0∈/S xϵ1i1 xϵ2i2 · · · xϵ3i3 . (3.9) Now we are interested in extracting a basis F from F. The elements of F will be the (P, ω)-partition enumerators of a particular type of chains, which w…
Figure 12
Figure 12. Figure 12: Example of a signed graph −→Σ , its corresponding symmetric graph G−→Σ and the Hasse diagram of the associated poset Pτ . • The (Σ, π)-rank of m ∈ {±1, . . . , ±d} is the length ℓ of the longest subword πi1 πi2 · · · πiℓ of π−d · · · π−1π1 · · · πd := π(−d)· · · π(−1)…
Figure 13
Figure 13. Figure 13: −→Σ 5,2 and −→Σ ν 5,2 . Proof. Say e = {i, j}, then {xi = −xj} ∈ HΣ, and this hyperplane separates the points having all positive coordinates from those having all negative coordinates. As the orien￾tation in −→Σ is acyclic there exists a chamber C corresponding to th…
Figure 14
Figure 14. Figure 14: Example of a directed path with an even number of vertices for i = 4. that keeps the number of ascents invariant. Also note that we can decompose (1, −1) = (0, 1)(0, −1)(0, 1) so by applying this method 3 times we can define Φ0 satisfying the required conditions. □ 2 …
Figure 15
Figure 15. Figure 15: Example of the involution Φ3 where the switched vertex is circled and the figure shows the coloring of −→Σ 9,3. The edges correspond￾ing to ascents are drawn in orange. The corresponding monomials are t 15x0x2x 2 3x 2 −3x 2 4x−4 and t 15x1x2x 2 3x−3x 2 4x 2 −4 respect…
Figure 16
Figure 16. Figure 16: Two examples of directed signed graphs −→Σ such that X−→Σ (x;t) ∈/ SSym. [ADM22] J.C. Aval, K. Djenabou and P.R. McNamara, Quasisymmetric functions distinguishing trees. J. Algebr. Comb. 6 (2023) 595-614. [BS98] N. Bergeron and F. Sottile, Schubert polynomials, the Br…

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