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Tree Descent Polynomials: Unimodality and Central Limit Theorem

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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.

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Acknowledgements

SP was partially supported by NSF-DMS 1815832.

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Correspondence to Svetlana Poznanović.

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Grady, A., Poznanović, S. Tree Descent Polynomials: Unimodality and Central Limit Theorem. Ann. Comb. (2020). https://doi.org/10.1007/s00026-019-00484-1

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