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Tree Descent Polynomials: Unimodality and Central Limit Theorem
Annals of Combinatorics ( IF 0.5 ) Pub Date : 2020-01-16 , DOI: 10.1007/s00026-019-00484-1
Amy Grady , Svetlana Poznanović

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.

中文翻译:

树后裔多项式:单峰和中心极限定理

对于其哈塞图是有根的平面森林F的位姿,我们考虑相应的树后裔多项式\(A_F(q)\),这是F的后裔数量的生成函数。当森林是一条路径时,\(A_F(q)\)专用于经典的欧拉多项式。我们证明\(A_F(q)\)的系数序列是单峰的,并且如果\(\ {T_ {n} \} \)\(| T_ {n} | = n \)的树的序列并且最大下降度\(D_ {n} = O(n ^ {0.5- \ epsilon}),\),则标注\(T_ {n} \)的下降次数渐近是正常的。
更新日期:2020-01-16
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