Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Tree Descent Polynomials: Unimodality and Central Limit Theorem


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3


  1. 1.

    A. Björner and M.L. Wachs. q-Hook length formulas for forests. J. Combin. Theory Ser. A, 52(2):165 – 187, 1989.

  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.

  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.

  4. 4.

    D. Foata and D. Zeilberger. Graphical major indices. J. Comput. Math., 68(1):79 – 101, 1996.

  5. 5.

    G. Frobenius. Uber die Bernoullischen und die Eulerschen Polynome. Sitzungsberichte der Preussische Akademie der Wissenschaften, 809–847, 1910.

  6. 6.

    V. Gasharov. On the Neggers–Stanley conjecture and the Eulerian polynomials. J. Combin. Theory Ser. A, 82(2):134 – 146, 1998.

  7. 7.

    Ira Gessel. Counting forests by descents and leaves. The Electronic Journal of Combinatorics, 3(2), Research paper #5, 1996.

  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.

  9. 9.

    J. W. Moon. On the maximum degree in a random tree. Michigan Math. J., 15(4):429–432, 1968.

  10. 10.

    John Shareshian and Michelle L Wachs. Chromatic quasisymmetric functions. Adv. Math., 295:497–551, 2016.

  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.

Download references


SP was partially supported by NSF-DMS 1815832.

Author information

Correspondence to Svetlana Poznanović.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Grady, A., Poznanović, S. Tree Descent Polynomials: Unimodality and Central Limit Theorem. Ann. Comb. (2020). https://doi.org/10.1007/s00026-019-00484-1

Download citation