A modular relation for the chromatic symmetric functions of (3+1)-free posets
By Mathieu Guay-Paquet. 2013
119 citations recorded by the source corpus.
We consider a linear relation which expresses Stanley's chromatic symmetric function for a poset in terms of the chromatic symmetric functions of some closely related posets, which we call the modular law. By applying this in the context of (3+1)-free posets, we are able to reduce Stanley and Stembridge's conjecture that the chromatic symmetric functions of all (3+1)-free posets are e-positive to the case of (3+1)-and-(2+2)-free posets, also known as unit interval orders. In fact, our reduction can be pushed further to a much smaller class of posets, for which we have no satisfying characterization. We also obtain a new proof of the fact that all 3-free posets have e-positive chromatic symmetric functions.
Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/55e9f3d06495a2e1ee690ad62be80743979ee93b
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Citations in the corpus
- Recent Trends in Quasisymmetric Functions (2018, background): “Guay-Paquet [48] proved that the chromatic symmetric function of a (3+1)-free poset is a convex combination of chromatic symmetric functions of posets which are both (3+1)-free and (2+2)-free.”
- Unit Interval Orders and the Dot Action on the Cohomology of Regular Semisimple Hessenberg Varieties (2015, unknown): “Firstly, Guay-Paquet posted a proof of Conjecture 3 which is completely independent of and, in many ways, complementary to our proof [21]”
- The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture (2017, background): “jecture) Let Γ = (V,E) be the incomparability graph of a unit interval order. Then XΓ(x) is e-positive. (In fact, the Stanley-Stembridge conjecture is stated more generally, but Guay-Paquet showed in [9] that the above special case implies the general version.) Shareshian and Wachs linked the Stanle”