Combinatorics·OpenAI Math·Result 169·Lean formalisation listed
Shareshian–Wachs elementary positivity
One of the 372 results in OpenAI's public mathematics catalogue, filed under combinatorics: 1 preprint, the latest dated 24 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a result titled ‘Shareshian–Wachs elementary positivity’ (combinatorics) in a preprint dated 24 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More, is on YouTube.
Chapter 22 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 7:17 · watch from there on YouTube →
What OpenAI claims
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]. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity. (Lean)
Paper
Elementary positivity of chromatic quasisymmetric functions
OpenAI's abstract We prove that the chromatic quasisymmetric function of every natural unit interval graph is elementary-positive over ℕ[q]. This resolves the elementary-positivity part of the Shareshian–Wachs conjecture.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 24 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- 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]. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity. (Lean)
- Where is the proof?
- In the paper, Elementary positivity of chromatic quasisymmetric functions (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 22).
Also in
- Chapter 7 of 10 in Borsuk's Conjecture Fails in Dimension 9, Sidorenko's Conjecture and 8 More · starts at 2:06 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · Compilation · 29:10 · chapter 22 from 7:17 · description, sources and chapters →
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