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On Enumeration of Families of Genus Zero Permutations

  • Sen-Peng Eu
  • Tung-Shan Fu
  • Yeh-Jong Pan
  • Chien-Tai TingEmail author
Original paper
  • 49 Downloads

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.

Keywords

Genus zero permutation Noncrossing partition André permutation Simsun permutation Smooth permutation Sign-balance identity 

Notes

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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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsNational Taiwan Normal UniversityTaipeiTaiwan ROC
  2. 2.Chinese Air Force AcademyKaohsiungTaiwan ROC
  3. 3.Department of Applied MathematicsNational Pingtung UniversityPingtungTaiwan ROC
  4. 4.School of Mathematics and StatisticsZhaoqing UniversityZhaoqingChina
  5. 5.Center for General Education, Chinese Air Force AcademyKaohsiungTaiwan ROC

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