Stanley–Stembridge e-positivity conjecture

Establish that the incomparability graph of every (3+1)-free poset is e-positive, meaning that its chromatic symmetric function has a nonnegative expansion in the elementary symmetric-function basis.

Background

The paper identifies the Stanley–Stembridge conjecture as a longstanding problem concerning chromatic symmetric functions. It asserts e-positivity for incomparability graphs of (3+1)-free posets and is presented as a major context for the study of e-positivity in graph classes, including trees.

References

The famous Stanley-Stembridge conjecture states that the incomparability graph of each (3 + 1)-free poset is e-positive, i.e., a non-negative linear expansion of elementary symmetric functions, which is equivalent to a conjecture on immanants of Jacobi-Trudi matrices [23].

— A quantitative way to e-positivity of trees  (2503.09484 - Li, 12 Mar 2025) in Section 1, Introduction, page 1

The famous Stanley-Stembridge conjecture states that the incomparability graph of each (3 + 1)-free poset is e-positive, i.e., a non-negative linear expansion of elementary symmetric functions, which is equivalent to a conjecture on immanants of Jacobi-Trudi matrices [23].

— A quantitative way to e-positivity of trees  (2503.09484 - Li, 12 Mar 2025) in Section 1, Introduction, page 1

The famous Stanley-Stembridge conjecture states that the incomparability graph of each (3 + 1)-free poset is e-positive, i.e., a non-negative linear expansion of elementary symmetric functions, which is equivalent to a conjecture on immanants of Jacobi-Trudi matrices [23].

— A quantitative way to e-positivity of trees  (2503.09484 - Li, 12 Mar 2025) in Section 1, Introduction, page 1