Elementary positivity of chromatic quasisymmetric functions
Source: github.com/openai/math/tree/main/preprints/Elementary-Posit…
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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I'll remark on some problems in algebraic combinatorics. Combinatorial invariance of KL polynomials . Proved true, I'd say in agreement with the majority's sentiment. Shareshian--Wachs conjecture. Proved true, in line with nearly universal expectations. Saxl's conjecture. Proved true, in line with…
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Answer to: Status of the Stanley–Stembridge conjecturemathoverflow.netJoshua P. SwansonNov 13, 2024
Regarding question (1), I coded Hikita's (6) this evening. It checks out through $n \leq 7$ and, for connected graphs, $n=8$ . Here's my Jupyter notebook , which I ran with SageMath 10.2.rc4. Personally I'm convinced. I don't see how Hikita's (6) could possibly fail for higher $n$ for this…