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#Chromatic symmetric functions in the elementary basis

Let \(G\) be a graph and let \(\chrom_G(\xvec)\) denote its chromatic symmetric function. An \(\elementaryE\)-expansion is an expression \[\chrom_G(\xvec) = \sum_{\lambda} c_\lambda \elementaryE_\lambda(\xvec).\] We say that \(\chrom_G(\xvec)\) is \(\elementaryE\)-positive if all \(c_\lambda\) are non-negative integers. If \(G\) is naturally labeled and \(\chrom_G(\xvec;q)\) is symmetric, then the stronger \(q\)-refined question asks whether \[\chrom_G(\xvec;q) = \sum_{\lambda} c_\lambda(q) \elementaryE_\lambda(\xvec) \qquad\text{with}\qquad c_\lambda(q) \in \setN[q].\]

The \(q\)-refined question is only available in families where \(\chrom_G(\xvec;q)\) is symmetric, most importantly for unit interval graphs or circular unit arc digraphs. For general graphs, the default question is the specialization \(q=1.\)

#The two main conjectures

The Stanley–Stembridge conjecture was formulated by R. Stanley and J. Stembridge [SS93], and then in the language of chromatic symmetric functions by Stanley [Sta95].

Theorem (Hikita, [Hik24]).

Let \(P\) be a \((3+1)\)-free poset. Then the chromatic symmetric function of the incomparability graph of \(P\) is \(\elementaryE\)-positive.

By work of M. Guay-Paquet [Gua13], it is enough to prove the result for unit interval graphs. Hikita’s proof gives a formula for the elementary coefficients of the Shareshian–Wachs \(q\)-chromatic refinement, and this formula specializes positively at \(q=1.\) Thus the theorem settles the Stanley–Stembridge conjecture, but it does not settle the following refinement.

Hikita later introduced \((q,t)\)-chromatic symmetric functions for unit interval graphs using level-one polynomial representations of affine Hecke algebras of type \(A\) [Hik25]. These functions refine the chromatic quasisymmetric functions in the sense that Hikita recovers the Shareshian–Wachs function from the \((q,t)\)-theory [Thm. 6.1, Hik25]. The specialization at \(q=\infty\) gives the probabilistic interpretation of the elementary coefficients used in Hikita’s proof of the Stanley–Stembridge conjecture. J. Huh, B.-H. Hwang, D. Kim, J. S. Kim, and J. Oh refine Hikita’s theorem using the \(g_{\mathfrak{m},k}(\xvec;q)\)-functions of A. Abreu and A. Nigro [HHKK+25]. They prove the \(\elementaryE\)-positivity of \(g_{\mathfrak{m},k}(\xvec;1),\) give a Schur expansion for a related weighted sum in terms of \(P\)-tableaux with \(1\) in the upper-left corner, and introduce a restricted modular law which determines such functions from their values on disjoint unions of paths.

Conjecture (Shareshian–Wachs, [Conj. 4.9, SW12]).

If \(G\) is a unit interval graph with its natural labeling, then \[\chrom_G(\xvec;q) = \sum_{\lambda} c_\lambda(q)\elementaryE_\lambda(\xvec) \qquad\text{has}\qquad c_\lambda(q) \in \setN[q].\]

A desirable combinatorial proof would give a statistic \(\mu\) on acyclic orientations of \(G\) such that \[\chrom_G(\xvec;q) = \sum_{\theta \in AO(G)} q^{\asc(\theta)} \elementaryE_{\mu(\theta)}(\xvec).\] This is the form in which one can see both the \(q\)-grading and the elementary-basis positivity at the same time. Gergely Berczi and Jonas Kluver use reinforcement learning to propose a universal counting formula for coefficients of chromatic symmetric functions of unit interval graphs [BK24]. Their conjectural formula counts disjoint tuples of Eschers satisfying graph-independent concatenation conditions. The conjecture extends to proper circular unit arc digraphs; see [Ell17, AP18].

Conjecture (See [Conj. 5.1, SW16]).

Write \[\chrom_G(\xvec;q) = \sum_{j=0}^m q^j a_j(\xvec),\] where \(m\) is the number of edges of a unit interval graph \(G.\) Then \(a_{j+1}(\xvec)-a_j(\xvec)\) is \(\elementaryE\)-positive for all \(0 \leq j \lt (m-1)/2.\)

This unimodality conjecture also extends to the circular unit arc digraph setting.

#Kinds of elementary expansions

It is useful to distinguish the following kinds of results.

  • A combinatorial formula gives the coefficient of \(\elementaryE_\lambda\) as the cardinality or weight-enumerator of a set of objects. This is the strongest kind of evidence for an eventual Shareshian–Wachs proof.

  • A sign-reversing involution starts from a signed \(\elementaryE\)-expansion and cancels all negative terms. Several proofs for special unit interval families have this form.

  • A recursive or generating-function proof shows that the chromatic symmetric functions satisfy an \(\elementaryE\)-positive recurrence with \(\elementaryE\)-positive initial data. This proves positivity, but does not always identify a natural coefficient statistic.

  • A noncommutative proof establishes the stronger \((\elementaryE)\)-positivity of the chromatic symmetric function in noncommuting variables, as in [GS01]. This implies ordinary \(\elementaryE\)-positivity.

  • A negative result usually means that one graph in the stated family has a negative elementary coefficient. It does not mean that every graph in that family is non-positive.

#Chronological guide

The following list is a compact guide to the main families and methods.

  • 1993–1995. Stanley and Stembridge introduce the conjectural positivity problem [SS93]. Stanley proves \(\elementaryE\)-positivity for paths and cycles, and obtains positive formulas for complete graphs and co-triangle-free graphs [Sta95].

  • 2001. D. Gebhard and B. Sagan introduce a chromatic symmetric function in noncommuting variables and prove noncommutative \((\elementaryE)\)-positivity for \(K_\alpha\)-chains and for diamond and path chains [GS01].

  • 2012–2016. J. Shareshian and M. Wachs introduce the \(q\)-refinement and the Shareshian–Wachs conjecture [SW12, SW16]. The \(q\)-refinement is symmetric for unit interval graphs and specializes to the ordinary chromatic symmetric function at \(q=1.\)

  • 2017–2022. Several special cases for unit interval graphs are proved: triangular ladders [Dah18], lollipops and lariats [DW18], the Abelian and bounce-number-two cases [HP19, CH19, NT22, LS22], and melting lollipop families [HNY20].

  • 2018–2021. For graph classes outside the unit interval setting, positive and negative families are sorted in work on \(H\)-free graphs [Tsu18, HHT19, FHM19], on claw-contractible-free graphs [DFW20], and on \((claw,2K_2)\)-free graphs [LY21].

  • 2021–2023. Certain cycle-chord and tadpole-type families are treated using noncommutative and recursive methods [WW22]. F. Tom gives a signed \(\elementaryE\)-expansion which yields new positive graph families [Tom25]. Coefficient-level positivity for partitions with at most two parts is proved in [AN23, RS23].

  • 2024–2026. Recent results include the \(2+1+1\)-avoiding unit interval orders [MPW24], twinned paths and cycles [BCCC+25], all cycle-chord graphs [Wan25], clocks [CHW26], conjoined graphs [QTW25], gluing graphs at a single vertex [TV26], adjacent cycle-chains [TV26], twinned lollipops and kayak paddle graphs [TW24], and refinements of Hikita’s proof [HHKK+25, GMRW+25]. Hook-shape immanant characters can be written as non-negative sums of Stanley–Stembridge characters by N. R. T. Lesnevich [Les24]; this proves a hook-shape case of a conjecture of R. Stanley and J. Stembridge on immanant characters.

#Unit interval and \(q\)-refined results

When the graph is a unit interval graph, one should record whether the result proves the full \(q\)-refined statement or only the specialization \(q=1.\) The following families are among the main cases where the refinement is known or where the proof is naturally stated in the chromatic quasisymmetric setting.

  • Complete graphs, paths, cycles, and the directed-cycle analogues have explicit formulas in the Shareshian–Wachs/Ellzey setting [SW12, Ell17, AP18].

  • If both a unit interval graph and its complement are unit interval graphs, then \(\chrom_G(\xvec)\) is \(\elementaryE\)-positive [FHM19].

  • For Abelian area sequences and related bounce-number-two families, the literature contains recursive, cohomological, and sign-reversing-involution proofs [HP19, CH19, NT22, LS22].

  • For area sequences with bounce number three, some coefficients are treated in [CH19], and this is extended in [Wan22].

  • All elementary coefficients indexed by partitions with at most two parts are non-negative, by [AN23] using the cohomology of Hessenberg varieties, and independently by [RS23].

  • The \(2+1+1\)-avoiding unit interval orders, equivalently the area sequences with \(i-2 \leq a_i,\) are treated in [MPW24] using strand diagrams and representation theory.

#Further positive families

The following examples are useful landmarks for the ordinary \(q=1\) problem.

  • Noncommutative \((\elementaryE)\)-positive families include \(K_\alpha\)-chains, diamond and path chains, and several later families proved by adapting the Gebhard–Sagan framework [GS01, WW22, WZ24].

  • Lollipops, lariats, triangular ladders, and generalized pyramid graphs give some of the standard small graph families where direct signed expansions can be made positive [Dah18, DW18, LY21].

  • Twinning preserves \(\elementaryE\)-positivity for paths and cycles, with both positive generating functions and positive recurrences [BCCC+25]. Twinning does not preserve \(\elementaryE\)-positivity for arbitrary graphs [LLWY21].

  • Cycle-chord graphs are \(\elementaryE\)-positive [WW22, Wan25]. The proof of the general case uses the composition method of [WZ24].

  • Graphs obtained by gluing at a single vertex give a flexible source of new examples. In particular, gluing sequences of unit interval graphs and cycles gives \(\elementaryE\)-positive graphs [TV26].

  • Adjacent cycle-chains are \(\elementaryE\)-positive, and the same methods extend to graphs formed by connecting a sequence of cycles and cliques [TV26].

  • Positive \(\elementaryE\)-expansions are known for KPKP graphs, twinned lollipops, and kayak paddle graphs [TW24]. This refines the earlier \(\elementaryE\)-positivity of kayak paddle graphs from [AWVW24].

  • Several tree families are now classified. For example, [WW23] classifies positivity for all broom graphs and most double broom graphs, and [TWW24] proves \(\elementaryE\)-positivity for the spiders \(S(4m+2,2m,1).\)

#Negative results and obstructions

Negative results are important: they indicate which graph-theoretic hypotheses are doing real work.

  • The star \(K_{1,3}\) is already not \(\elementaryE\)-positive. More generally, large-degree cut vertices and several families of trees obstruct \(\elementaryE\)-positivity [DSVW20].

  • The saltire, augmented saltire, and triangular tower families give negative examples in the study of claw-contractible-free graphs [DFW20].

  • For \(H\)-free graph classes, some families are positive and some contain counterexamples. See [HHT19, FHM19] for a systematic treatment.

  • Some spider and related families are not \(\elementaryE\)-positive; see [FKKM+20] and the later work on trees and connected partitions [Tom26].

  • Tom proves that trees with a vertex of degree at least \(5,\) trees with a degree-\(4\) vertex not adjacent to a leaf, and four-legged spiders are not \(\elementaryE\)-positive [Tom26]. A quantitative connected-partition approach to further tree obstructions is developed in [Li25].

  • Twinning an \(\elementaryE\)-positive graph at a vertex can destroy even Schur positivity, and hence can destroy \(\elementaryE\)-positivity [LLWY21].

Example (The smallest star obstruction).

For the star graph \(K_{1,3},\) \[\chrom_{K_{1,3}}(\xvec) = \elementaryE_{211} + 5\elementaryE_{31} -2\elementaryE_{22} + 4\elementaryE_4.\] The coefficient of \(\elementaryE_{22}\) is negative, so this graph is not \(\elementaryE\)-positive.

#Coefficient-level tests

Several papers study individual elementary coefficients rather than a whole family of graphs. For example, [CHL23] gives explicit conditions for when a coefficient in the \(\elementaryE\)-expansion is positive, and [CZ22] studies elementary-basis coefficients in relation to acyclic orientations and sinks. The change-of-basis perspective in [ST26] gives another way to turn information about particular partitions into graph-theoretic positivity statements.

After Hikita’s proof, several papers have focused on making the coefficients more explicit. The Macdonald expansion of [GMRW+25] rederives Hikita’s formula, while [Sie25] studies upper and lower combinatorial bounds for elementary coefficients. Finally, [Kra26] proves that the partitions which always have non-negative elementary coefficients for every finite graph are precisely the hook partitions.

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