Closed forms for derangement numbers in terms of the Hessenberg determinants

Original Paper


In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.


Derangement number Closed form Hessenberg determinant Tridiagonal determinant Generating function Recurrence relation Derivative 

Mathematics Subject Classification

05A05 05A10 05A15 11B37 11B39 11B83 11B65 11B83 11C08 11C20 11Y55 15A15 65F40 



The second author was partially supported by China Postdoctoral Science Foundation with Grant No. 2015M582619. The authors are grateful to the anonymous referees for their careful corrections to and valuable comments on the original version of this paper.


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Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  1. 1.Institute of MathematicsHenan Polytechnic UniversityJiaozuoChina
  2. 2.College of MathematicsInner Mongolia University for NationalitiesTongliaoChina
  3. 3.Department of Mathematics, College of ScienceTianjin Polytechnic UniversityTianjinChina
  4. 4.Department of Mathematics and InformaticsWeinan Normal UniversityWeinanChina
  5. 5.School of Mathematics and InformaticsHenan Polytechnic UniversityJiaozuoChina

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