Weighted quasisymmetric enumerator for generalized permutohedra

  • Vladimir GrujićEmail author
  • Marko Pešović
  • Tanja Stojadinović


We introduce a weighted quasisymmetric enumerator function associated with generalized permutohedra. It refines the Billera, Jia and Reiner quasisymmetric function which also includes the Stanley chromatic symmetric function. Besides that, it carries information of face numbers of generalized permutohedra. We consider more systematically the cases of nestohedra and matroid base polytopes.


Generalized permutohedron Quasisymmetric function Combinatorial Hopf algebra f-polynomial 

Mathematics Subject Classification

Primary 05E05 Secondary 52B05 16T05 



  1. 1.
    Aguiar, M., Ardila, F.: Hopf monoids and generalized permutohedra, arXiv:1709.07504
  2. 2.
    Aguiar, M., Bergeron, N., Sottile, F.: Combinatorial Hopf algebras and generalized Dehn–Sommerville relations. Compos. Math. 142, 1–30 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ardila, F., Klivans, C.: The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory Ser. B 96, 38–49 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Billera, L., Jia, N., Reiner, V.: A quasisymmetric function for matroids. European J. Combin. 30, 1727–1757 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carr, M.P., Devadoss, S.L.: Coxeter complexes and graph-associahedra. Topology Appl. 153(12), 2155–2168 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Crapo, H., Schmitt, W.: A free subalgebra of the algebra of matroids. European J. Combin. 26, 1066–1085 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Crapo, H., Schmitt, W.: A unique factorization theorem for matroids. J. Combin. Theory Ser. A 112, 222–249 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Feichtner, E.M., Sturmfels, B.: Matroid polytopes, nested sets and Bergman fans. Port. Math. 62(4), 437–468 (2005)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gel’fand, I.M., Serganova, V.V.: Combinatorial geometries and torus strata on homogeneous compact manifolds. Russian Math. Surveys 42(2), 133–168 (1987)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gel’fand, I.M., Goresky, M., MacPherson, R., Serganova, V.: Combinatorial geometries, convex polyhedra, and Schubert cells. Adv. Math. 63, 301–316 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grinberg, D., Reiner, V.: Hopf algebras in combinatorics, arXiv:1409.8356
  12. 12.
    Grujić, V.: Quasisymmetric functions for nestohedra. SIAM J. Discrete Math. 31(4), 2570–2585 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grujić, V.: Counting faces of graphical zonotopes. Ars Math. Contemp. 13, 227–234 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Grujić, V., Stojadinović, T.: Counting faces of nestohedra. Sem. Lothar. Combin. 78B, 17 (2017)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Humpert, B., Martin, J.: The incidence Hopf algebra of graphs. SIAM J. Discrete Math. 26(2), 555–570 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Oxley, J.G.: Matroid theory. Oxford University Press, New York (1992)zbMATHGoogle Scholar
  17. 17.
    Postnikov, A.: Permutohedra, associahedra, and beyond. Int. Math. Res. Not. IMRN 2009, 1026–1106 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Postnikov, A., Reiner, V., Williams, L.: Faces of generalized permutohedra. Doc. Math. 13, 207–273 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Schmitt, W.R.: Incidence Hopf algebras. J. Pure Appl. Algebra 96(3), 299–330 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Stanley, R.P.: A symmetric function generalization of the chromatic polynomial of a graph. Adv. Math. 111, 166–194 (1995)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsUniversity of BelgradeBelgradeSerbia
  2. 2.Faculty of Civil EngineeringUniversity of BelgradeBelgradeSerbia

Personalised recommendations