Tree Descent Polynomials: Unimodality and Central Limit Theorem

TL;DR

This paper studies the properties of tree descent polynomials associated with rooted plane forests, proving their unimodality and establishing a central limit theorem for the distribution of descents in large trees.

Contribution

It introduces the concept of tree descent polynomials for rooted plane forests and proves their coefficient unimodality and asymptotic normality under certain conditions.

Findings

Coefficient sequence of $A_F(q)$ is unimodal.

Number of descents in large trees follows a normal distribution.

Special case reduces to classical Eulerian polynomial.

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