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Decomposition and Merging Algorithms for Noncrossing Forests

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Abstract

A noncrossing forest is a forest drawn on n points numbered in counterclockwise order on a circle in such a way that its edges are rectilinear and do not cross. Utilizing analytic combinatorics, Flajolet and Noy obtained the number of noncrossing forests with n vertices and k components. In this paper, we will give a new representation for noncrossing forests. Based on such respresentation, we establish a decomposition algorithm and a merging algorithm, which leads to a bijection between labeled noncrossing forests and sets of rooted matches. As an application, we derive a new formula, which is a refinement of the formula given by Flajolet and Noy.

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Acknowledgements

We are grateful to the referees for valuable suggestions. This work was supported by the National Natural Science Foundation of China, the Natural Science Foundation of Hebei Province, the Top Young-aged Talents Program of Hebei Province and the One-Hundred Outstanding Innovative Talents Scheme of the Hebei Province Education Department.

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Authors and Affiliations

  1. School 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 & Xiaoming Deng

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  1. Sabrina X. M. Pang
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  2. Lun Lv
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  3. Xiaoming Deng
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Correspondence to Lun Lv.

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Pang, S.X.M., Lv, L. & Deng, X. Decomposition and Merging Algorithms for Noncrossing Forests. Graphs and Combinatorics 38, 2 (2022). https://doi.org/10.1007/s00373-021-02415-5

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  • DOI: https://doi.org/10.1007/s00373-021-02415-5

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