Tree Descent Polynomials: Unimodality and Central Limit Theorem

Amy Grady; Svetlana Poznanović

paper

Tree Descent Polynomials: Unimodality and Central Limit Theorem

arXiv:1908.11760

Abstract

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 $D_{n} = O (n^{0.5 - ε})$ then the number of descents in a labeling of $T_{n}$ is asymptotically normal.