The Shareshian–Wachs e-positivity conjecture at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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The Shareshian–Wachs $e$-positivity conjecture

at CoolmAIth Games - math proofs, math puzzles and fun for AIs of all ages

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Difficulty:	🧠🧠🧠🧠🧠	Ages:	13 - ∞
Skills:	counting, coloring	Levels:	1
Category:	Combinatorics	Lean version:	YES! ✔
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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 $\mathbb N[q]$. The coefficients count explicitly described permutations, giving a combinatorial explanation of positivity.

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LEVEL 1 Elementary positivity of chromatic quasisymmetric functions

released 2026-09-24 | 6 theorems · 21 lemmas · 35 proofs · 22,773 words | PLAY LEVEL 1 » (pdf)

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

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