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On Enumeration of Families of Genus Zero Permutations

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Abstract

The genus of a permutation \(\sigma \) of length n is the nonnegative integer \(g_{\sigma }\) given by \(n+1-2g_{\sigma }={\textsf {cyc}}(\sigma )+{\textsf {cyc}}(\sigma ^{-1}\zeta _n)\), where \({\textsf {cyc}}(\sigma )\) is the number of cycles of \(\sigma \) and \(\zeta _n\) is the cyclic permutation \((1,2,\ldots ,n)\). On the basis of a connection between genus zero permutations and noncrossing partitions, we enumerate the genus zero permutations with various restrictions, including André permutations, simsun permutations, and smooth permutations. Moreover, we present refined sign-balance results on genus zero permutations and their analogues restricted to connected permutations.

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Acknowledgements

The authors thank the referees for carefully reading the manuscript and providing helpful suggestions. This research is partially supported by Ministry of Science and Technology (MOST), Taiwan, under Grants 107-2115-M-003-009-MY3 (S.-P. Eu), 107-2115-M-153-003-MY2 (T.-S. Fu), and 108-2115-M-013-001 (C.-T. Ting).

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Authors and Affiliations

  1. Department of Mathematics, National Taiwan Normal University, Taipei, 11677, Taiwan ROC

    Sen-Peng Eu

  2. Chinese Air Force Academy, Kaohsiung, 82047, Taiwan ROC

    Sen-Peng Eu

  3. Department of Applied Mathematics, National Pingtung University, Pingtung, 90003, Taiwan ROC

    Tung-Shan Fu

  4. School of Mathematics and Statistics, Zhaoqing University, Zhaoqing, 526061, China

    Yeh-Jong Pan

  5. Center for General Education, Chinese Air Force Academy, Kaohsiung, 82047, Taiwan ROC

    Chien-Tai Ting

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  1. Sen-Peng Eu
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  2. Tung-Shan Fu
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  4. Chien-Tai Ting
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Correspondence to Chien-Tai Ting.

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Eu, SP., Fu, TS., Pan, YJ. et al. On Enumeration of Families of Genus Zero Permutations. Graphs and Combinatorics 35, 1337–1360 (2019). https://doi.org/10.1007/s00373-019-02111-5

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  • DOI: https://doi.org/10.1007/s00373-019-02111-5

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