Result 169, Combinatorics

Shareshian–Wachs elementary positivity

Resolves the elementary-positivity part of the Shareshian–Wachs conjecture: the chromatic quasisymmetric function of every natural unit interval graph has elementary-basis coefficients in N[q]\mathbb N[q]. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity.

Lean formalization Proof

The bigger picture

Why it matters

For graphs formed by overlapping equal-length intervals on a line, a complicated coloring formula is claimed to have a positive counting explanation. This turns an algebraic sign question into a question about finite collections of permutations.

What changes?

The unreviewed manuscript reports elementary positivity for every natural unit interval graph: an interval-overlap graph with equal-length intervals and vertices labeled in left-to-right order. Its chromatic quasisymmetric function records colorings in which adjacent vertices have different colors; q tracks how colors respect vertex order. The claim is that expansion into elementary symmetric functions, standard building blocks using products of distinct color variables, has coefficients that are polynomials in q with nonnegative integer coefficients. This resolves the elementary-positivity part of the Shareshian-Wachs conjecture.

What does that help mathematicians do?

The supplied summary says these coefficients count explicitly described permutations. That gives a combinatorial reason why negative coefficients cannot occur, rather than merely asserting an algebraic inequality. Setting q to one also yields elementary positivity for the corresponding ordinary chromatic symmetric function. Researchers can therefore deduce positivity both for the refined coloring invariant and for the version that forgets the vertex-order statistic.

Are there practical applications?

The immediate value is foundational: it connects graph colorings, symmetric-function expansions and permutation enumeration within this specific graph class. The counting interpretation offers concrete objects to study when investigating individual coefficients. The supplied sources do not establish a practical application, an efficient computation method, or elementary positivity for arbitrary graphs.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Elementary positivity of chromatic quasisymmetric functions

September 24, 2026 51 pages

We prove that the chromatic quasisymmetric function of every natural unit interval graph is elementary-positive over N[q]\mathbb N[q]. This resolves the elementary-positivity part of the Shareshian–Wachs conjecture.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026,
  author = {{OpenAI}},
  title = {{Elementary positivity of chromatic quasisymmetric functions}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/paper.pdf}{OAI:Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/169.md.

Shareshian–Wachs elementary positivity

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The elementary-positivity part of the Shareshian–Wachs conjecture asks whether the chromatic quasisymmetric function of every natural unit interval graph has nonnegative coefficients in the elementary basis. The formalization proves this over N[q]\mathbb N[q]. It constructs an explicit elementary-basis expansion indexed by the graph's permitted nondescent permutations, with each term weighted by a nonnegative power of qq determined by its graph inversions. The expansion holds for every finite number of color variables.

Comparator links

Result Comparator statement
Elementary positivity for natural unit interval graphs ElementaryPositivity.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.