Tree Descent Polynomials: Unimodality and Central Limit Theorem

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.