Elementary positivity of chromatic quasisymmetric functions

Follow-up works

• Per Alexandersson, Lilan Dai, Flip classes and elementary expansions of vertical-strip LLT polynomials (arXiv:2610.26294, 19 Oct 2026). Proves elementary positivity of flip-class sums for vertical-strip LLT polynomials; its introduction cites this paper's Theorem 1.1 as having proved that the elementary coefficients of the chromatic quasisymmetric function of every natural unit-interval graph lie in ℕ[q].

Comments and discussion

• Joshua P. Swanson, Status of the Stanley–Stembridge conjecture (MathOverflow answer, update 2026-10-07). Swanson discusses OpenAI's Shareshian–Wachs paper, describes its exposition as opaque, and says its approach appears independent of Hikita's proof and the Griffin–Mellit–Romero–Weigl–Wen proof.

Formalization

• OpenAI, Shareshian–Wachs elementary positivity: Lean scope note (OpenAI mathematics release, 2026-10-06). The scope note reports a formal elementary expansion over $\mathbb N[q]$, indexed by permitted nondescent permutations and weighted by graph inversions, for every finite number of color variables, and links the comparator statement.
• OpenAI, lean/ComparatorChallenges/ElementaryPositivity.lean (openai/math, 2026). Comparator statement: elementaryPositivityWitness returns a PermutationWitness (a partition theta σ per nondescent permutation and the expansion identity in MvPolynomial (Fin r) (Polynomial ℕ) for every r); the body is left as sorry for the comparator.
• OpenAI, lean/OAI/Combinatorics/Chromatic/Main.lean (openai/math, 2026). The release's construction of elementaryPositivityWitness, built from a finite-search table (selectedTable, exists_successful_table) in the ElementaryPositivity namespace.

Related prior literature mentioned by others

• Tatsuyuki Hikita, A proof of the Stanley–Stembridge conjecture (arXiv:2410.12758, 2024-10-16; revised 2025-12-25). Swanson's discussion contrasts OpenAI's approach with Hikita's probabilistic formula, which establishes the Stanley–Stembridge specialization at $q=1$.
• Sean T. Griffin, Anton Mellit, Marino Romero, Kevin Weigl and Joshua Jeishing Wen, On Macdonald expansions of $q$-chromatic symmetric functions and the Stanley–Stembridge Conjecture (arXiv:2504.06936, 2025-04-09). This Macdonald-expansion proof rederives Hikita's formula and is the approach from which Swanson says OpenAI's argument appears independent.

Possibly relevant literature (found by the agent, not checked)

• Foster Tom, A signed $e$-expansion of the chromatic quasisymmetric function (arXiv:2311.08020, 2023-11-14; revised 2024-05-08). Its signed elementary expansion for all natural unit interval graphs and sign-reversing involution for $K$-chains provide a comparison for the paper's positive permutation-indexed formula (Theorem 1.1) and the finite cancellation construction in Section 6.
• Tatsuyuki Hikita, (q,t)-chromatic symmetric functions (arXiv:2503.23597, 2025). Gives a probabilistic interpretation of the e-expansion coefficients used in Hikita's Stanley–Stembridge proof; relevant to the paper's discussion of Hikita's rational q-weights versus coefficientwise ℕ[q]-positivity.
• Soojin Cho, Seonjeong Park, Permutation module decomposition of the cohomology of Hessenberg varieties associated with lollipop graphs (arXiv:2607.03284, 3 Jul 2026). Constructs a permutation-module decomposition of Hessenberg cohomology (a Cho–Hong–Lee conjecture) for lollipop graphs; related to the paper's Corollary 1.2, which gives only an abstract isomorphism to sums of Young permutation modules.
• Boris Kafidov, Log-concavity of elementary coefficients for low-rank abelian Hessenberg graphs, with a counterexample in general (arXiv:2607.20595, Jul 2026). Studies unimodality/log-concavity of the elementary coefficients c_λ(q), the part of the Shareshian–Wachs conjecture the paper explicitly does not address.

Digests and expositions

• SymCat (symmetricfunctions.com), Chromatic symmetric functions in the elementary basis (page dated 2026-10-07). Discusses the paper's Theorem 1.1 and Section 6, explaining that its permutation statistic uses auxiliary geometric choices and ordered finite matchings, and notes that the separate elementary-unimodality conjecture remains open.