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A Refinement of Leaves on Noncrossing Trees

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Abstract

A leaf of a NC-tree is called a left leaf if it has a label less than that of its parent, and it is called a right leaf otherwise. We enumerate NC-trees with given number of left leaves and right leaves. As an application, the combinatorial interpretation of three identities is obtained. By extending a well-known involution on plane trees to NC-trees, we derive that the two parameters, the number of left leaves and the number of non-root proper vertices, have a symmetric joint distribution in the NC-trees. Finally, a bijection between right leaves and NC-forests is also presented.

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Authors and Affiliations

  1. College of Mathematics and Statistics, Hebei University of Economics and Business, Shijiazhuang, 050061, People’s Republic of China

    Sabrina X. M. Pang

  2. School of Sciences, Hebei University of Science and Technology, Shijiazhuang, 050018, People’s Republic of China

    Lun Lv

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  1. Sabrina X. M. Pang
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  2. Lun Lv
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Correspondence to Lun Lv.

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Pang, S.X.M., Lv, L. A Refinement of Leaves on Noncrossing Trees. Graphs and Combinatorics 29, 131–143 (2013). https://doi.org/10.1007/s00373-011-1097-z

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  • DOI: https://doi.org/10.1007/s00373-011-1097-z

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