Tree Descent Polynomials: Unimodality and Central Limit Theorem

  • Amy Grady
  • Svetlana PoznanovićEmail author


For a poset whose Hasse diagram is a rooted plane forest F, we consider the corresponding tree descent polynomial \(A_F(q)\), which is a generating function of the number of descents of the labelings of F. When the forest is a path, \(A_F(q)\) specializes to the classical Eulerian polynomial. We prove that the coefficient sequence of \(A_F(q)\) is unimodal and that if \(\{T_{n}\}\) is a sequence of trees with \(|T_{n}| = n\) and maximal down degree \(D_{n} = O(n^{0.5-\epsilon }),\) then the number of descents in a labeling of \(T_{n}\) is asymptotically normal.



SP was partially supported by NSF-DMS 1815832.


  1. 1.
    A. Björner and M.L. Wachs. q-Hook length formulas for forests. J. Combin. Theory Ser. A, 52(2):165 – 187, 1989.MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Bóna and R. Ehrenborg. A combinatorial proof of the log-concavity of the numbers of permutations with k runs. J. Combin. Theory Ser. A, 90(2):293 – 303, 2000.MathSciNetCrossRefGoogle Scholar
  3. 3.
    González D’León and S Rafael. A Note on the \(\gamma \)-coefficients of the tree Eulerian polynomial. Electron. J. Comb., 23(1):P1.20, 2016.MathSciNetzbMATHGoogle Scholar
  4. 4.
    D. Foata and D. Zeilberger. Graphical major indices. J. Comput. Math., 68(1):79 – 101, 1996.MathSciNetCrossRefGoogle Scholar
  5. 5.
    G. Frobenius. Uber die Bernoullischen und die Eulerschen Polynome. Sitzungsberichte der Preussische Akademie der Wissenschaften, 809–847, 1910.Google Scholar
  6. 6.
    V. Gasharov. On the Neggers–Stanley conjecture and the Eulerian polynomials. J. Combin. Theory Ser. A, 82(2):134 – 146, 1998.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ira Gessel. Counting forests by descents and leaves. The Electronic Journal of Combinatorics, 3(2), Research paper #5, 1996.Google Scholar
  8. 8.
    S. Janson. Normal convergence by higher semiinvariants with applications to sums of dependent random variables and random graphs. Ann. Probab., 16(1):305–312, 1988.MathSciNetCrossRefGoogle Scholar
  9. 9.
    J. W. Moon. On the maximum degree in a random tree. Michigan Math. J., 15(4):429–432, 1968.MathSciNetCrossRefGoogle Scholar
  10. 10.
    John Shareshian and Michelle L Wachs. Chromatic quasisymmetric functions. Adv. Math., 295:497–551, 2016.MathSciNetCrossRefGoogle Scholar
  11. 11.
    R.P. Stanley. Log-concave and unimodal sequences in algebra, combinatorics, and geometry. Ann. N.Y. Acad. Sci., 576(1):500–535, 1989.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of Mathematical and Statistical SciencesClemson UniversityClemsonUSA

Personalised recommendations