Tree Descent Polynomials: Unimodality and Central Limit Theorem

2019/08/30 by Grady, Amy, Poznanović, Svetlana
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/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.

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