The Shareshian-Wachs conjecture asserts elementary positivity, together with unimodality, for chromatic quasisymmetric functions of natural unit interval graphs. Such a graph on is specified by weakly increasing integers , ..., with , using edges for . Its elementary-basis expansion is

where ranges over the partitions of and , with an elementary symmetric polynomial. Elementary positivity requires each polynomial to have nonnegative integer coefficients. Unimodality further requires each coefficient sequence to weakly increase and then weakly decrease, making it a unimodal sequence.

OpenAI (2026) announced coefficientwise elementary positivity for every natural unit interval graph. The reported positivity holds coefficientwise in , which is stronger than positivity after setting . Liu (2026) subsequently reported the stronger decomposition

where is the number of edges, denotes the floor function, and the are nonnegative integers. This gives elementary positivity and unimodality for every natural unit interval graph and every partition of its number of vertices. Liu's argument uses the framework in OpenAI's manuscript rather than deriving unimodality from positivity alone. OpenAI reports an AI-generated proof. Liu reports GPT-6 Pro and Claude Fable assistance with mathematical exploration, proof development, and checking. As of Nov. 3, 2026, independent specialist verification of the complete claimed proofs had not been reported.

See also

Chromatic Quasisymmetric Function, Elementary Symmetric Polynomial, Unimodal Sequence

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References

Liu, M. "Elementary Unimodality for Natural Unit Interval Graphs." Preprint. Oct. 22, 2026. https://doi.org/10.5281/zenodo.23573763.OpenAI. "Elementary Positivity of Chromatic Quasisymmetric Functions." 24 Sep 2026. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Elementary-Positivity-of-Chromatic-Quasisymmetric-Functions-September-24-2026/paper.pdf.Shareshian, J. and Wachs, M. L. "Chromatic Quasisymmetric Functions." Adv. Math. 295, 497-551, 2016. https://doi.org/10.1016/j.aim.2015.12.018.

Cite this as:

Weisstein, Eric W. "Shareshian-Wachs Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Shareshian-WachsConjecture.html

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