The symmetric functions catalog

#Kromatic symmetric functions

A \(K\)-theoretical analog of the chromatic symmetric functions was introduced in [CPS26], called the kromatic symmetric function. It serves as a non-homogeneous lift of the chromatic symmetric functions introduced by R. Stanley [Sta95]. One main feature is that these expand positively into Grothendieck symmetric functions whenever the graph is a claw-free incomparability graph of a poset.

E. Marberg gives a Hopf-algebraic construction of the kromatic symmetric function and proves that it has a positive expansion into multifundamental quasisymmetric functions [Cor. 3.25, Mar25]. He also introduces two quasisymmetric \(q\)-analogues. One is multifundamental-positive but is symmetric only for cluster graphs [Thms. 4.7 and 4.8, Mar25]; the other has a positive expansion into stable Grothendieck functions when the graph is the incomparability graph of a natural unit interval order [Thm. 4.21, Mar25].

L. Pierson gives an explicit expansion of the kromatic symmetric function in the \(K\)-analogue of the power-sum basis and proves integrality and a sign rule for the coefficients [Thm. 1.1 and Cor. 1.5, Pie26]. A later refinement gives a Lyndon-heap interpretation of these coefficients and proves that the \(\omega\)-image is positive in the same \(K\)-power-sum basis [Thms. 1.2 and 1.3, Pie25].

The kromatic symmetric function contains more information than the ordinary chromatic symmetric function. In particular, knowing it is equivalent to knowing the multiset of independence polynomials of all induced subgraphs [Cor. 1.6, Pie25], and it determines the number of induced copies of several small independence-unique graphs [Pie24]. However, it does not distinguish all graphs: L. Pierson and S. Samanta construct four pairs of non-isomorphic graphs on eight vertices with the same kromatic symmetric function [PS26].

Bibliography

1. [CPS26]Logan Crew, Oliver Pechenik and Sophie Spirkl. The Kromatic symmetric function: A K-theoretic analog of ${X}_{G}$. Canadian Mathematical Bulletin:1–23, 2026.
   .bib

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   @article{CrewPechenikSpirkl2023x,
     author = {Crew, Logan and Pechenik, Oliver and Spirkl, Sophie},
     title = {The {K}romatic symmetric function: {A} {K}-theoretic analog of ${X}_{G}$},
     year = {2026},
     journal = {Canadian Mathematical Bulletin},
     pages = {1--23},
     publisher = {Canadian Mathematical Society},
     doi = {10.4153/s0008439526101830},
     url = {http://dx.doi.org/10.4153/s0008439526101830},
     issn = {1496-4287}
   }
   

2. [Mar25]Eric Marberg. Kromatic Quasisymmetric Functions. The Electronic Journal of Combinatorics, 32(1):P1.11, 2025.
   .bib

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   @article{Marberg2025,
     author = {Marberg, Eric},
     title = {Kromatic {Q}uasisymmetric {F}unctions},
     year = {2025},
     journal = {The Electronic Journal of Combinatorics},
     volume = {32},
     number = {1},
     pages = {P1.11},
     publisher = {The Electronic Journal of Combinatorics},
     doi = {10.37236/13207},
     url = {http://dx.doi.org/10.37236/13207},
     issn = {1077-8926}
   }
   

3. [Pie24]Laura Pierson. Counting induced subgraphs with the Kromatic symmetric function. arXiv:2403.15929, 2024.
   .bib

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   @article{Pierson2024x,
   Author = {Laura Pierson},
   Title = {Counting induced subgraphs with the {K}romatic symmetric function},
   Year = {2024},
   Eprint = {2403.15929},
     url = {https://arxiv.org/abs/2403.15929},
   journal = {arXiv e-prints}
   }
   

4. [Pie25]Laura Pierson. Power sum expansions for Kromatic symmetric functions using Lyndon heaps. Annals of Combinatorics, 2025.
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   @article{Pierson2025,
     author = {Laura Pierson},
     title = {Power sum expansions for {K}romatic symmetric functions using {L}yndon heaps},
     year = {2025},
     journal = {Annals of Combinatorics},
     doi = {10.1007/s00026-025-00785-8},
     url = {https://doi.org/10.1007/s00026-025-00785-8}
   }
   

5. [Pie26]Laura Pierson. A power sum expansion for the Kromatic symmetric function. Discrete Mathematics, 349(5):114957, 2026.
   .bib

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   @article{Pierson2026,
     author = {Pierson, Laura},
     title = {A power sum expansion for the {K}romatic symmetric function},
     year = {2026},
     journal = {Discrete Mathematics},
     volume = {349},
     number = {5},
     pages = {114957},
     publisher = {Elsevier BV},
     doi = {10.1016/j.disc.2025.114957},
     url = {http://dx.doi.org/10.1016/j.disc.2025.114957},
     issn = {0012-365X}
   }
   

6. [PS26]Laura Pierson and Soham Samanta. On graphs with equal and different Kromatic symmetric functions. Discrete Applied Mathematics, 387:137–159, 2026.
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   @article{PiersonSamanta2026,
     author = {Pierson, Laura and Samanta, Soham},
     title = {On graphs with equal and different {K}romatic symmetric functions},
     year = {2026},
     journal = {Discrete Applied Mathematics},
     volume = {387},
     pages = {137--159},
     publisher = {Elsevier BV},
     doi = {10.1016/j.dam.2026.02.039},
     url = {http://dx.doi.org/10.1016/j.dam.2026.02.039},
     issn = {0166-218X}
   }
   

7. [Sta95]Richard P. Stanley. A symmetric function generalization of the chromatic polynomial of a graph. Advances in Mathematics, 111(1):166–194, 1995.
   .bib

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   @article{Stanley1995,
   author = {Richard P. Stanley},
   title = {A Symmetric Function Generalization of the Chromatic Polynomial of a Graph},
   journal = {Advances in Mathematics},
   volume = {111},
   number = {1},
   pages = {166--194},
   year = {1995},
   issn = {0001-8708},
   doi = {10.1006/aima.1995.1020},
   url2 = {http://www.sciencedirect.com/science/article/pii/S0001870885710201}
   }