Chromatic quasisymmetric functions

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#Chromatic quasisymmetric functions

G. Arunkumar relates Stanley’s chromatic symmetric functions to Borcherds algebras [Aru21]. Another Hopf-algebraic viewpoint is given by J.-C. Aval, N. Bergeron, and J. Machacek, who introduce invariants of permutations, orders, and graphs connected with the nabla operator [ABM20]. J. A. White defines chromatic quasisymmetric class functions for digraphs and, more generally, for combinatorial Hopf monoids [Whi21, Whi25]; these variants incorporate group actions into the chromatic quasisymmetric setting.

#Historical overview

R. Stanley introduced the chromatic symmetric function of a graph $G$ in [Sta95]. This definition was later refined by J. Shareshian and M. Wachs in [SW12] and [SW16], where the graph $G$ is labeled and a parameter $q$ is introduced.

In 2016, B. Ellzey [Ell17] presented a generalization where labeled graphs are replaced with oriented graphs. This model is equivalent to the Shareshian–Wachs model when the orientation of the edges has no directed cycle. See also P. Alexandersson and G. Panova [AP18], who considered the same model from a different perspective.

Positivity in the elementary symmetric function basis has been a central problem for chromatic symmetric functions and their $q$-refinement [SW12, SW16]. The corresponding conjectures are the Stanley–Stembridge conjecture and the Shareshian–Wachs conjecture, respectively. Shareshian and Wachs also conjectured that $\omega \chrom_G(\xvec;q)$ is related to the equivariant cohomology of regular semisimple Hessenberg varieties.

In [Hwa22], B.-H. Hwang introduces an additional refinement of the chromatic quasisymmetric functions. He defines a flip operator on colorings and shows that connected components under such flips are Schur-positive. The ascent parameter introduced by Shareshian–Wachs is preserved by such flips. It is conjectured that each such component is positive in the elementary symmetric function basis. See also [Hwa23] for a connection with noncommutative symmetric functions. A proof for Schur positivity of the flip-components is also given in [BEPS22].

For a representation-theoretical perspective, see [Ska20], where connections with immanents and characters are studied.

Y. Nishimura proves that the second Kneser chromatic function $X_{K_{\setN,2}}(G)$ is a complete invariant for trees [Nis26]. This fits into the Kneser chromatic functions of T. Hasebe, H. Miezaki, A. Munemasa, Y. Nishimura, Y. Sakuma, and S. Tsujie, where $X_{K_{\setN,1}}(G)$ is Stanley’s chromatic symmetric function and the whole family is a complete invariant for finite graphs.

L. Crew, E. Haithcock, J. Reynes, and S. Spirkl introduce a chromatic multisymmetric function for graphs whose vertex set is partitioned into parts [CHRS24]. They use homogeneous sets and homogeneous pairs to turn known relations for ordinary chromatic symmetric functions into new ones, including a generalization of Guay-Paquet’s reduction of the Stanley–Stembridge conjecture to unit interval graphs.

F. Aliniaeifard, V. Wang, and S. v. Willigenburg define a centered chromatic symmetric function $y_{G:v}$ using a quotient of noncommutative symmetric functions [AWVW24]. This viewpoint gives new $e$-positive unit interval graph families, operations preserving the stronger centered positivity property, and noncommutative refinements of sink-counting theorems for acyclic orientations. J. M. Campbell gives another noncommutative direction: a lift of Stanley’s chromatic symmetric function to $\mathrm{NSym}$ [Cam24]. The lift is indexed by directed graphs and specializes under the commutative image to the ordinary chromatic symmetric function of the underlying graph.

R. S. G. D’León and M. L. Wachs construct a different chromatic symmetric analogue from multiweighted bond posets [DW26]. Its one-variable specialization is the reversed chromatic polynomial, while its top homogeneous component is a new Möbius symmetric function of the graph.

#Definition

Let $G$ be an oriented graph with $n$ vertices. Given a vertex-coloring $\kappa:G\to \setP,$ let $\asc(\kappa)$ denote the number of directed edges $(u,v)$ in $G$ such that $\kappa(u) \lt \kappa(v).$

The chromatic quasisymmetric function of $G$ is defined as \[\chrom_G(\xvec;q) \coloneqq \sum_{\substack{\kappa : G\to \setP \\ \kappa \text{ proper}} } x_{\kappa(1)}\dotsm x_{\kappa(n)} q^{\asc(\kappa)},\] where the sum runs over all proper colorings of the graph $G.$ A proper coloring is a coloring in which no edge has endpoints of the same color. The function $\chrom_G(\xvec;q)$ is quasi-symmetric in $\xvec.$

From the definition, it is clear that \[\chrom_G(1^k;1)\] is the number of ways to color $G$ with $k$ colors, so the classical chromatic polynomial $\chi_G(k)$ is given by the specialization $\chrom_G(1^k;1).$ See [SV21] for a nice article with bijective proofs of properties of $\chi_G(k).$ Z. Cheng, Z. Lei, Y. Wang, and Y. Zhang construct a cohomology theory for the signed chromatic polynomial of a signed graph [CLWZ22]. Its graded Euler characteristic is the signed chromatic polynomial, and its long exact sequence categorifies signed deletion–contraction. J.-C. Aval and R. Melgar introduce chromatic quasisymmetric functions for signed graphs [AM25]. Their construction is guided by Zaslavsky’s signed-graph hyperplane arrangements and leads to an algebra of signed quasisymmetric functions.

Note that every labeling induces an orientation, by orienting edges toward the larger label. This labeled model is what was introduced in [SW12, SW16].

#Claw-free graphs

R. Stanley recorded the conjecture, credited to V. Gasharov, that $\chrom_G(\xvec;1)$ is Schur-positive whenever $G$ is claw-free [Sta98]. This conjecture is false. J. P. Matherne and Alejandro H. Morales give two line graphs whose chromatic symmetric functions have negative Schur coefficients [MM26]. Namely, let $G_i=L(H_i),$ where \[\begin{aligned} E(H_1)=\{&(1,8),(2,7),(3,6),(3,10),(4,5),(4,9),\\ &(5,9),(6,10),(7,9),(7,10),(8,9),(8,10)\}, \end{aligned}\] and \[\begin{aligned} E(H_2)=\{&(1,8),(2,5),(2,10),(3,7),(3,10),(4,6),\\ &(4,9),(5,9),(6,9),(7,10),(8,9),(8,10)\}. \end{aligned}\] Then $G_1$ and $G_2$ have $12$ vertices, with $22$ and $21$ edges, respectively, and the relevant Schur coefficients are \[[\schurS_{3333}]\chrom_{G_1}(\xvec;1)=-64, \qquad [\schurS_{3333}]\chrom_{G_2}(\xvec;1)=-40.\] Since every line graph is claw-free, these are counterexamples to the general conjecture.

Independently, J. Prajapati gives a short computation by hand for the counterexample with coefficient $-64,$ verifies it by exact implementations, and reports an exhaustive census showing that every connected claw-free graph on at most $11$ vertices is Schur-positive [Pra26]. The same census finds exactly two non-Schur-positive isomorphism classes on $12$ vertices, with the two coefficients $-64$ and $-40$ above. D. G. L. Wang, K. Zhang, and T. Y. Zhao combine this census with further exact computations and construct an infinite family of connected line graphs whose chromatic symmetric functions are not Schur-positive [WZZ26].

Positive results remain for special claw-free families. Using an involution inspired by the slinky rule, the authors of [SW24] prove Schur positivity for the family of graphs called generalized nets.

R. Davila, H. Schuerger, and B. Small characterize claw-free graphs using zero forcing invariants [DSS24]. They prove that $G$ is claw-free if and only if every induced subgraph $H$ satisfies $Z(H)=Z_+(H).$

E. Y. H. Li, G. M. X. Li, A. L. B. Yang, and Z.-X. Zhang introduce a strengthening of the nice property, called the strongly nice property [LLYZ24]. They prove that Schur positivity implies strong niceness, and that the graphs whose induced subgraphs are all strongly nice are precisely the claw-free graphs. Thus hereditary strong niceness captures the claw-free condition, but the line-graph counterexamples above show that it is not sufficient for Schur positivity.

#Products of chains

Distributive lattices give a separate family of negative tests when a poset is viewed through its incomparability graph. G. M. X. Li, D. Qiu, A. L. B. Yang, and Z.-X. Zhang construct distributive lattices that are not nice and hence not Schur-positive, and also distributive lattices that are nice but still not Schur-positive [LQYZ24]. Here, $\mathbf{r}$ denotes a chain with $r$ elements. Later, D. G. L. Wang and K. Zhang prove that the product lattices $\mathbf{m}\times\mathbf{2}$ and $\mathbf{m}\times\mathbf{3}$ are Schur-positive exactly for $m\leq7$ and $m\leq6,$ respectively. They also show that $\mathbf{m}\times\mathbf{3}$ is not strongly nice for $m\geq44$ [WZ25].

K. Zhang supplies a negative coefficient throughout the remaining large-$m$ range. Put $m=n+k$ and \[\rho=(2n+k-1,2n+k-3,\dotsc,k+5,k,k,4).\]

Theorem (Zhang, [Zha26, Thm. 3.1]).

If $n\geq4$ and $k\geq2n-1,$ then \[[\schurS_\rho]\, \chrom_{\operatorname{inc}(\mathbf{m}\times\mathbf{n})}(\xvec;1)\lt{}0.\] In particular, $\mathbf{m}\times\mathbf{n}$ is not Schur-positive whenever $n\geq4$ and $m\geq3n-1.$

Together with the range $n+5\leq m\leq3n-2$ proved in [LQYZ24], this shows that $\mathbf{m}\times\mathbf{n}$ is not Schur-positive for every $n\geq3$ and $m\geq n+5$; for $n=2,$ nonpositivity holds for $m\geq8$ [Zha26, Cor. 3.4].

#Unit interval graphs

The following definition using area sequences is from [AP18].

A circular unit arc digraph (or circular Dyck path) $\Gamma_\avec$ is a directed graph with vertex set $[n]$ and edges \[\begin{aligned} (i-a_i) \to i, \; (i-a_i + 1) \to i, \; (i-a_i + 2) \to i, \; \dotsc \; , (i-1) \to i % i \to i+1, \; i \to i+2,\; \dotsc, i \to i+a_i \end{aligned}\] for all $i = 1,\dotsc,n,$ where indices are taken modulo $n,$ and the integers $\avec=(a_1,\dotsc,a_n)$ satisfy

$0 \leq a_i \leq n-1$ for $1 \leq i \leq n,$
$a_{i+1} \leq a_{i} + 1$ for $1 \leq i \leq n,$ (index mod $n$).

Whenever $a_1=0,$ $\Gamma_\avec$ is a unit interval graph. The sequence $\avec = (a_1,a_2,\dotsc,a_n)$ is called the area sequence of the graph.

There are Catalan $\frac{1}{n+1}\binom{2n}{n}$ many unit interval graphs on $n$ vertices, and a total of \[(n+2)\binom{2n-1}{n-1} - 2^{2n-1}\] circular unit arc digraphs on $n$ vertices, see [ALP19] for a proof. This sequence appears as A194460 in the OEIS.

Example

The area sequences $(0,1,2,1,1)$ and $(3,2,1,2,2)$ can be presented as Dyck diagrams, or in the second circular case, a circular Dyck diagram:

$\square $ $ \square $ $ \square $ $ 5$ $ \square $ $ \square $ $ 4$ $ 3$ $ 2$ $ 1$ $ \square $ $ \square $ $ 5$ $ \square $ $ \square $ $ 4$ $ \square $ $ \square $ $ \square $ $ 3$ $ \square $ $ \square $ $ 2$ $ \square $ $ 1$

The labeled boxes represent vertices, the empty boxes are edges (connecting the vertex in the same column with the vertex in the same row), and the squares represent non-edges. The first graph has 5 edges, while the second has 9 edges. Note that circular Dyck diagrams wrap around at top and bottom. Edges are directed upwards (or rightwards) in these diagrams.

The Abelian unit interval graphs on vertices $\{1,2,\dotsc,n\}$ are the unit interval graphs where there is no vertex $j$ such that both $(1,j)$ and $(j,n)$ are non-edges. The number of Dyck paths of size $n$ with an Abelian unit interval graph is $2^{n-1}.$

#Properties of chromatic quasisymmetric functions

#Symmetry

Whenever $G$ is a unit arc digraph as above, $\chrom_G(\xvec;q)$ is symmetric for labeled graphs [SW12, SW16] and for oriented graphs [Ell17, AP18]. Furthermore, the coefficients in $\setZ[q]$ in the monomial basis are palindromic. To be precise, \[\chrom_\avec(\xvec;q)=\sum_\lambda c_\lambda(q) \monomial_\lambda(\xvec) \qquad \Longrightarrow \qquad c_\lambda(q) = q^{|\avec|}c_\lambda(1/q),\] where $|\avec|$ denotes the sum $a_1+a_2+\dotsb + a_n.$

M. Gillespie, J. Pappe, and K. Salois study when a chromatic quasisymmetric function is symmetric [GPS24]. They show that symmetry of a product of quasisymmetric functions forces each factor to be symmetric, reduce the problem to connected graphs, and prove that directed trees other than directed paths have nonsymmetric chromatic quasisymmetric functions. L. Colmenarejo and I. Klein introduce total chromatic quasisymmetric functions of graphs [CK26]. One variant normalizes over all vertex labelings, while another is built from acyclic orientations; the paper compares their monomial-quasisymmetric expansions and gives explicit formulas for star graphs.

#Reciprocity

Let $\chi_G(k)$ be the chromatic polynomial associated with the graph $G$ on the vertex set $[n].$ O. Bernardi and P. Nadeau [BN20] gave the following interpretation of the specialization $(-1)^{n-i}\chi^{(i)}_G(-j).$

Theorem

Let $i,j \geq 0.$ Then $(-1)^{n-i}\chi^{(i)}_G(-j)$ counts the number of tuples \[\left( (V_1, \theta_1),\dotsc,(V_{i+j}, \theta_{i+j}) \right)\] such that

$V_1,\dotsc,V_{i+j}$ is a set-partition of the vertices of $G,$
$\theta_\ell$ is an acyclic orientation of the graph induced by $V_\ell,$
for all $\ell \in [i],$ $V_\ell$ is non-empty and $\theta$ has a unique source, given by $\min V_\ell.$

#Relationship with LLT polynomials

Carlsson–Mellit [CM17, Prop. 3.4] proved the following plethystic relationship between chromatic quasisymmetric functions and unicellular LLT polynomials indexed by unit interval graphs on $n$ vertices: \[(q-1)^{-n} \LLT_\avec[\xvec(q-1);q] = \chrom_\avec(\xvec;q).\] This relationship is not always true in the circular case, that is, for area sequences where $a_1 \gt 0.$ The relationship is reminiscent of the technique introduced by J. Haglund, M. Haiman and N. Loehr [HHL05].

This plethystic relationship is generalized in [NT19, Thm. 6.2], where a noncommutative lift is given. J.-C. Novelli and J.-Y. Thibon also introduce a noncommutative lift of the unicellular LLT polynomials.

M. D’Adderio, R. Riccardi, and V. Siconolfi extend this picture from Dyck graphs to interval graphs [DRS23]. They introduce quasisymmetric functions attached to increasing spanning forests and use them to give a forest expansion for chromatic quasisymmetric functions of interval graphs [DRS23, Thm. 7.2]. They also define LLT quasisymmetric functions for ordered simple graphs, recovering the unicellular LLT symmetric functions for Dyck graphs [DRS23, Thm. 7.7], and prove an interval-graph generalization of the Carlsson–Mellit plethystic identity [DRS23, Thm. 10.1].

#Fundamental quasisymmetric positivity

The standard $P$-partition expansion gives \[\chrom_G(\xvec;q) = \sum_{\theta \in AO(G)} q^{\asc(\theta)} K_{P(\theta)}(\xvec)\] where $AO(G)$ denotes the set of acyclic orientations of $G,$ and $K_{P(\theta)}(\xvec)$ is the generating function of strict order-preserving maps from the poset $P(\theta)$ given by the transitive closure of the directed edges in $\theta,$ to the set of positive integers.

From this, it follows that $\chrom_G(\xvec;q)$ is positive in the fundamental quasisymmetric basis.

V. Tewari, A. Wilson, and P. Zhang study a nonsymmetric refinement for Dyck graphs and prove positivity in the fundamental slide basis [TWZ22].

#The $w$-polynomial and $X$-descents

Let $G$ be a graph on $n\geq1$ vertices. Applying the fundamental-length specialization to its chromatic symmetric function gives \[w_G(t) \coloneqq \Psi_t\bigl(\chrom_G(\xvec)\bigr) =(1-t)^{n+1}\sum_{m\geq0}\chi_G(m)t^m.\] This is F. Brenti’s $w$-polynomial [Bre92, Prop. 4.1]. In particular, this specialization of $\chrom_G(\xvec)$ depends only on the ordinary chromatic polynomial.

More generally, let $R\subseteq[n]\times[n]$ be a relation and set \[\DES_R(\sigma) \coloneqq \{j\in[n-1]:(\sigma_j,\sigma_{j+1})\in R\}, \qquad A_R(t)\coloneqq\sum_{\sigma\in\symS_n}t^{|\DES_R(\sigma)|}.\] The Rédei–Berge symmetric function is \[U_R(\xvec) \coloneqq \sum_{\sigma\in\symS_n}\gessel_{n,\DES_R(\sigma)}(\xvec),\] see [GS23, Def. 1.15]. Hence \[\Psi_t(U_R)=tA_R(t).\]

Let $h:[n]\to[n]$ be a Hessenberg function, and let $H_h$ be the natural unit interval graph with an edge $\{i,j\}$ whenever $i\lt{}j\leq h(i).$ The descending relation of the corresponding natural unit interval order is \[R_h\coloneqq\{(i,j):i\gt{}h(j)\}.\] A position $j$ of $\sigma$ is an $X$-descent if $(\sigma_j,\sigma_{j+1})\in R_h$; write \[\operatorname{xdes}_h(\sigma) \coloneqq |\DES_{R_h}(\sigma)|.\] Also, let $\inv_{H_h}(\sigma)$ count the pairs $i\lt{}j$ such that $\sigma_i\gt{}\sigma_j$ and $\{\sigma_i,\sigma_j\}$ is an edge of $H_h.$

Theorem (Shareshian–Wachs, [SW16, Thm. 3.1]).

For every Hessenberg function $h:[n]\to[n],$ \[\Psi_t\bigl(\omega\chrom_{H_h}(\xvec;q)\bigr) =t\sum_{\sigma\in\symS_n} q^{\inv_{H_h}(\sigma)}t^{\operatorname{xdes}_h(\sigma)}.\]

Proof

Shareshian and Wachs expand $\omega\chrom_{H_h}(\xvec;q)$ in fundamental functions indexed by the $H_h$-inversion number and the descent set of the incomparability order. Their fundamental functions use decreasing indices. Reversing the corresponding compositions gives the increasing-index convention used here and preserves their lengths. Applying $\Psi_t$ gives the displayed identity.

At $q=1,$ the right-hand side is $tA_{R_h}(t).$ Since natural unit interval graphs are chordal, the chordal $w$-polynomial theorem shows that $A_{R_h}(t)$ has only real nonpositive zeros.

#Power-sum positivity

In [AS19], it is proved that $\omega \chrom_G(\xvec;q)$ is always $\qPsi$-positive, that is, positive in a quasisymmetric power-sum basis. Consequently, $\omega \chrom_G(\xvec;q)$ is $\powerSum$-positive whenever $\chrom_G(\xvec;q)$ is symmetric. This was proved earlier in [Ell17], and in the case of unit interval graphs $G$ in [Ath15]. A combinatorial formula in the unit-interval case was conjectured earlier by Shareshian–Wachs.

The following expansion is proved in [AS19].

Theorem (Alexandersson, Sulzgruber (2019)).

Let $G$ be a graph on $n$ vertices. The quasisymmetric power-sum expansion of $\omega \chrom_G(\xvec;q)$ is given by \[\omega \chrom_G(\xvec;q) = \sum_{\theta \in AO(G)} q^{\asc(\theta)} \sum_{f \in \opsurj^{\ast}(\theta)} \frac{\qPsi_{\type(f)}(\xvec)}{z_{\type(f)}}\] where the first sum ranges over acyclic orientations of $G.$ The set $\opsurj_{\alpha}^{\ast}(\theta)$ consists of order-preserving surjections $f$ from $\theta$ to some $[k],$ such that $\theta$ restricted to the vertices $f^{-1}(j)$ has a unique sink. The type of $f$ is the composition $\alpha$ given by $\alpha_j \coloneqq |f^{-1}(j)|.$

In particular, for $q=1,$ $\chrom_G(\xvec) = \chrom_G(\xvec;1)$ is symmetric. Thus, the formula above gives \[\begin{aligned} \omega \chrom_G(\xvec) &= \sum_{\theta \in AO(G)} \sum_{f \in \opsurj^{\ast}(\theta)} \frac{\powerSum_{\type(f)}(\xvec)}{z_{\type(f)}}. \end{aligned}\] This can further be simplified, giving \[\begin{aligned} \omega \chrom_G(\xvec) &=\sum_{\theta \in AO(G)} \powerSum_{\type(\theta)}(\xvec), \end{aligned}\] where $\type(\theta)$ is given by the source-components of $\theta,$ see [BN20]. This formula is very similar to the stronger result by C. A. Athanasiadis [Ath15], which treats the case with general $q.$

#Schur positivity

In [Gas96], V. Gasharov proved that the $\chrom_G(\xvec;1)$ are Schur-positive with an explicit expansion, whenever $G$ is the incomparability graph of a $3+1$-free poset. In particular, this implies the unit-interval case, as those are given as the incomparability graphs of $3+1$ and $2+2$-free posets. Gasharov’s proof uses a kind of sign-reversing involution.

Gasharov’s result was later extended to the Shareshian–Wachs setting (with a $q$-parameter), by using a different involution. That is, $\chrom_G(\xvec;q)$ is Schur-positive for unit-interval graphs $G,$ see [SW12, Thm. 6.3]. D. Kim and P. Pylyavskyy give a Robinson–Schensted correspondence for unit interval orders which refines the Schur expansion through dual equivalence graphs [KP21].

Example

We have the following Schur expansions of chromatic quasisymmetric functions, indexed by area sequences: \[\begin{aligned} \chrom_{000}(x;q)&=\schurS_{111}+2 \schurS_{21}+\schurS_{3} \\ \chrom_{001}(x;q)&=(1+q) \schurS_{111}+(1+q) \schurS_{21} \\ \chrom_{011}(x;q)&=(1+2 q+q^2) \schurS_{111}+q \schurS_{21} \\ \chrom_{111}(x;q)&=(3 q+3 q^2) \schurS_{111} \\ \chrom_{012}(x;q)&=(1+2 q+2 q^2+q^3) \schurS_{111} \\ \chrom_{112}(x;q)&=(q+4 q^2+q^3) \schurS_{111} \\ \chrom_{122}(x;q)&=(3 q^2+3 q^3) \schurS_{111} \\ \chrom_{222}(x;q)&=6 q^3 \schurS_{111} \end{aligned}\]

Problem

Prove that $\chrom_G(\xvec;q)$ is Schur-positive (with coefficients in $\setN[q]$) in the case of circular unit arc digraphs. This conjecture has been verified for all circular unit arc digraphs up to 10 vertices.

Find a bijective proof of Schur positivity (using crystals or a version of RSK). Gasharov’s result provides a hint on how a crystal graph structure might be defined.

A step in the direction of a crystal structure is taken in [Ehr22], but the crystals described there are not crystals in the usual sense.

In [WW20], the authors give a (signed) formula for the Schur coefficients of the chromatic symmetric function of any graph. They do this by inverting the matrix of Kostka coefficients. The approach is reminiscent of the transition from a fundamental quasisymmetric expansion to a Schur expansion via the slinky rule. The authors also characterize all Schur-positive complete tripartite graphs.

#Elementary symmetric expansion

The Stanley–Stembridge conjecture (1993) was first formulated in [SS93] and [Sta95] (for $q=1$). The Shareshian–Wachs conjecture is the refinement using the general $q$-parameter, presented in [SW12, Conj. 4.9]. Positivity in the elementary symmetric function basis (for $q=1$) was proved by T. Hikita [Hik24]. A September 2026 preprint by OpenAI announces a proof of the $q$-refined positivity for every naturally labeled unit interval graph [Ope26, Thm. 1.1]. The separate elementary-unimodality conjecture remains open. For a survey on expansions in the elementary symmetric function basis, see this page.

The hook filter on the Gessel fundamental basis is useful for extracting hook-shaped Schur coefficients from chromatic symmetric functions; it is one of the tools already present in Stanley’s original development [Sta95].

#Deletion contraction property

There is an extension to extended chromatic symmetric functions which admits a deletion-contraction recurrence.

R. Orellana and G. Scott prove a triangle-deletion recurrence and use it to construct and compare graphs with equal chromatic symmetric functions [OS14].

#Connection with rook placements

Chromatic quasisymmetric functions of unit interval graphs have a strong connection with rook placements. Acyclic orientations and rook placements are in bijection; see, for example, [AP18]. This connection is expanded upon in [CMP23]. In particular, some new results in the Abelian case using $q$-hit numbers are given.

#Alpha-chromatic symmetric functions

J. Haglund, J. Oh, and M. Yoo introduce $\alpha$-chromatic symmetric functions, which add a real parameter $\alpha$ to the chromatic quasisymmetric functions of natural unit interval graphs [HOY25]. These functions interpolate between chromatic symmetric functions and unicellular LLT polynomials: the coefficient of the first $q$-falling-factorial term recovers the chromatic symmetric function, while the large-$\alpha$ limit recovers the Schur coefficients of the unicellular LLT polynomial up to a $q$-factorial factor.

Their main results include a positive monomial formula in an $\alpha$-binomial basis [HOY25, Thm. 3.1], an expansion into a falling-factorial basis whose coefficients are sums of ordinary chromatic symmetric functions [HOY25, Thm. 4.4], and a rook-theoretic solution of the Garsia–Remmel $q$-hit problem for Ferrers boards associated with Dyck paths [HOY25, Cor. 6.3].

#Chromatic symmetric functions of trees

R. Stanley conjectures [Sta95] that the chromatic symmetric function of a tree uniquely determines the tree. In [WYZ23], trees with exactly two vertices of degree greater than $2$ are shown to be distinguished by their chromatic symmetric functions.

For tree chromatic symmetric functions in the star basis, see [AMOZ23, GOT24]. Trees of diameter less than $5$ can be reconstructed from their chromatic symmetric functions. Caterpillar graphs are known to be reconstructed from their CSF [LS19]. Certain $q$-caterpillar trees can be reconstructed from their CSF [ANVS23].

Another approach differentiates in the power-sum coordinates. Define \[\Phi_T \coloneqq \left.\frac{\partial X_T}{\partial\powerSum_1}\right|_{\powerSum_1=0}.\] A tree is proper if every non-leaf vertex is adjacent to a leaf. Contracting all leaf edges gives its weighted skeleton, whose vertex weights remember how many leaves were absorbed. S. A. Awan proves that $\Phi_T$ distinguishes every proper tree whose repeated skeleton weights occur only at skeleton leaves [Awa26, Thm. 1]. He also gives an equivalent reformulation of Stanley’s full tree question: if $C_q(T)$ is obtained by attaching $q-1$ leaves to every vertex, then for every fixed $q\geq2,$ one has [Awa26, Thm. 2] \[\Phi_{C_q(T)}=\Phi_{C_q(T')} \quad\Longleftrightarrow\quad X_T=X_{T'}.\]

N. A. Mobaraki, Y. Gerivani, and S. G. Nezhad develop a framework for calculating chromatic symmetric functions in forest bases [MGN24]. Their method recovers the equivalence of the chromatic symmetric function and the $U$-polynomial on forests, and gives formulas for extracting monomial coefficients from induced subgraphs. They also explain the algorithm of J. Aliste-Prieto, A. d. Mier, R. Orellana, and J. Zamora as a special case of their method.

In [HT17, Thm. 1.3], it is proved that ordered trees are distinguished by their order quasisymmetric function, where the rooted tree is viewed as a poset and colors increase away from the root. Both the weakly increasing and strictly increasing versions of the quasisymmetric function distinguish ordered trees. The work by T. Hasebe and S. Tsujie is continued in [Zho20], where one is interested in reconstructing the tree from the quasisymmetric function.

It is still an open problem (posed by Hasebe and Tsujie) to prove that the P-partition quasisymmetric function distinguishes posets whose Hasse diagrams are trees.

In [AS19], we conjecture that in the family of directed trees, the chromatic quasisymmetric function uniquely determines the directed tree. This conjecture would follow from the conjecture by Hasebe and Tsujie. See also [FKKA+21].

There is a variant of Stanley’s CSF defined for rooted trees, and this rooted version is shown to distinguish non-isomorphic trees, see [LW24].

Chromatic symmetric functions associated with trees are in general not Schur-positive, see the counterexamples presented in [RS20]. In fact, no tree on $20$ vertices with maximal degree $10$ is Schur-positive.

See also [AMWZ24] for polynomial invariants of trees.

#Other families of graphs

Expanding the CSF in different bases recovers various classical statistics of graphs; see [CC23]. For coefficients in the elementary basis in relation to sinks, see [CZ22].

A. Kirillov, G. Nenashev, B. Shapiro, and A. Vaintrob introduce the loopy polynomial $L_G,$ whose deletion–contraction recurrence retains a contracted edge as a loop [KNSV26]. The Tutte polynomial is a specialization of $L_G,$ while $L_G$ determines Stanley’s chromatic symmetric function for every graph [KNSV26, Cor. 2.6 and Thm. 5.4]. For loopless graphs it also determines the degree sequence, induced-edge profiles, and independence polynomial; for simple graphs it determines the clique and matching polynomials.

Separating component size from external activity gives a refined loopy polynomial equivalent to both the extended $U$-polynomial and the extended polychromate [KNSV26, Thm. 6.3]. The authors conjecture that the ordinary loopy and $U$-polynomials have the same distinguishing power on simple graphs and verify this through eleven vertices. The analogous statement fails for loopless multigraphs [KNSV26, Thm. 8.10].

J.-Y. Thibon and D. G. L. Wang prove that the chromatic symmetric functions of the spiders $S(a,2,1)$ and $S(a,4,1)$ are Schur-positive for every $a\geq1$ [TW26, Thms. 4.2 and 4.6]. Their proof lifts the path formula to noncommutative symmetric functions, expands it in ribbon functions, and translates the Schur coefficients into weighted Yamanouchi-word counts. Together with earlier elementary-positivity results, this gives the complete elementary- and Schur-positivity classification of both families.

E. Bajo, M. Beck, and A. R. Vindas-Meléndez define $q$-chromatic polynomials by weighting proper colorings with $q^{\sum_v \lambda_v c(v)}$ [BBV24]. These polynomials specialize to the ordinary chromatic polynomial at $q=1$ and, for $\lambda=(1,\dotsc,1),$ to a principal specialization of Stanley’s chromatic symmetric function. Their results connect this invariant with $q$-Ehrhart polynomials, order polytopes, and strengthened tree-distinguishing questions.

#Connection with Hessenberg varieties

In [SW12, Conj. 5.3], a connection with cohomology of regular semisimple Hessenberg varieties was conjectured. This conjecture was resolved in 2015, when P. Brosnan and T. Chow proved that for unit-interval graphs $\avec,$ the symmetric function $\omega \chrom_\avec(\xvec;q)$ is the Frobenius characteristic of a symmetric group action on the cohomology of regular semisimple Hessenberg varieties. This was also proved independently by Guay-Paquet [Gua16].

For more details on this symmetric group action, see the dot action page.

L. Gagnon gives a finite-group analogue of this result: inducing the trivial character from certain unipotent subgroups of $\GL_n(\mathbb{F}_q)$ realizes the chromatic quasisymmetric function, up to the expected scalar factor [Gag24, Thm. 3.1]. The same paper relates these character values to point counts of nilpotent Hessenberg varieties over finite fields, and realizes vertical-strip LLT polynomials as unipotent constituents of induced characters [Gag24, Thm. 5.1].

A. Abreu, A. Nigro, and S. Ram give a finite-field point-counting formula for arbitrary Hessenberg varieties [ANR24]. If $T$ is a linear operator on $\mathbb{F}_q^n,$ then the number of $\mathbb{F}_q$-rational points of the Hessenberg variety $H(m,T)$ is \[|H(m,T)| = \left\langle F_T(\xvec),\omega\chrom_{G(m)}(\xvec;q)\right\rangle,\] where $F_T(\xvec)$ is an invariant-flag generating function expressible using modified Hall–Littlewood polynomials [ANR24, Thm. 1.1]. For triangulable $T,$ this also gives Poincaré polynomials of the corresponding complex Hessenberg varieties [ANR24, Cor. 3.8].

A. Abreu and A. Nigro relate characters of Kazhdan–Lusztig elements of the Hecke algebra, including chromatic symmetric functions of indifference graphs, to intersection cohomology of parabolic Lusztig varieties [AN25]. The resulting decomposition through partial flag varieties points toward recursive approaches to Haiman’s positivity conjecture and the Stanley–Stembridge conjecture.

Y.-H. Kiem and D. Lee introduce generalized Hessenberg varieties and formulate a weighted-graph version of the Shareshian–Wachs conjecture [KL24]. They prove the generalized conjecture, so the corresponding weighted chromatic quasisymmetric functions have Schur coefficients that are unimodal polynomials with nonnegative coefficients [KL24, Thm. 4.6 and Cor. 4.8]. They also propose a generalized Stanley–Stembridge question for elementary positivity in this weighted setting.

#Other results related to chromatic symmetric functions

S. Y. Lin and L. Pierson study a different multiplicative Hopf-algebra structure on chromatic symmetric functions and show that graph complementation, for unweighted triangle-free graphs, corresponds naturally to a Hopf algebra operation [LP25].

Several recent variants move the chromatic symmetric function into larger ambient algebras. Campbell’s $\mathrm{NSym}$ lift records directed-edge order data before applying the commutative image [Cam24], while J. L. Martin and M. B. Trist define chromatic MacMahon symmetric functions for vertex-weighted graphs [MT25].

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.bib

@article{Zhou2020x,
Author = {Jeremy Zhou},
Title = {Reconstructing Rooted Trees From Their Strict Order Quasisymmetric Functions},
Year = {2020},
journal = {arXiv e-prints},
Eprint = {2008.00424}
}

The symmetric functions catalog