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On a closed form for derangement numbers: an elementary proof

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We provide a very short proof of a Qi-Zhao-Guo closed form for derangements numbers based on the determinants of certain Hessenberg matrices. The proof is grounded on a basic result on finite sequences and an inductive argument.

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References

  1. Anđelić, M., da Fonseca, C.M.: A short proof for a determinantal formula for generalized Fibonacci numbers. Matematiche (Catania) 74, 363–367 (2019)

  2. Barrios Rolania, D.: On the Darboux transform and the solutions of some integrable systems. RACSAM 113, 1359–1378 (2019)

    Article  MathSciNet  Google Scholar 

  3. Barrios Rolania, D., Garcia-Ardila, J.C.: Geronimus transformations for sequences of \(d\)-orthogonal polynomials. RACSAM 114 (2020), #26

  4. Costabile, F.A., Gualtieri, M.I., Napoli, A.: Polynomial sequences: elementary basic methods and application hints. A survey. RACSAM 113, 3829–3862 (2019)

    Article  MathSciNet  Google Scholar 

  5. da Fonseca, C.M.: An identity between the determinant and the permanent of Hessenberg type-matrices. Czechoslovak Math. J. 61, 917–921 (2011)

    Article  MathSciNet  Google Scholar 

  6. He, Y., Pan, J.: Some recursion formulae for the number of derangement and Bell numbers. J. Math. Res. Appl. 36, 15–22 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Janjić, M.: Determinants and recurrence sequences. J. Integer Seq. 15 (2012), Article 12.3.5 (2012)

  8. Kittappa, R.K.: A representation of the solution of the \(n\)th order linear difference equation with variable coefficients. Linear Algebra Appl. 193, 211–222 (1993)

    Article  MathSciNet  Google Scholar 

  9. Knuth, D.E.: The Art of Computer Programming, vol. 1, Fundamental Algorithms, Third Edition, Addison-Wesley, Boston, NJ, (1997)

  10. Liu, C.L.: Introduction to Combinatorial Mathematics. McGraw-Hill, New York (1968)

    MATH  Google Scholar 

  11. Merca, M.: A note on the determinant of a Toeplitz-Hessenberg matrix. Spec. Matrices 1, 10–16 (2013)

    MathSciNet  MATH  Google Scholar 

  12. Qi, F., Wang, J.L., Guo, B.N.: Closed forms for derangement numbers in terms of the Hessenberg determinants. RACSAM 112, 933–944 (2018)

    Article  MathSciNet  Google Scholar 

  13. Sloane, N.J.A.: The On-Line Encyclopedia of Integer Sequences, https://oeis.org/

  14. Stanley, R.P.: Enumerative Combinatorics, Vol.1, 2nd Edition, Cambridge Studies in Advanced Mathematics (Book 49), Cambridge University Press, (2011)

  15. Vein, R., Dale, P.: Determinants and Their Applications in Mathematical Physics, Applied Mathematical Sciences 134, Springer-Verlag, New York (1999)

  16. Verde-Star, L.: Polynomial sequences generated by infinite Hessenberg matrices. Spec. Matrices 5, 64–72 (2017)

    Article  MathSciNet  Google Scholar 

  17. Verde-Star, L.: Divided differences and combinatorial identities. Stud. Appl. Math. 85, 215–242 (1991)

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Kuwait College of Science and Technology, Doha District, Block 4, P.O. Box 27235, 13133, Safat, Kuwait

    Carlos M. da Fonseca

  2. University of Primorska, FAMNIT, Glagoljsaška 8, 6000, Koper, Slovenia

    Carlos M. da Fonseca

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  1. Carlos M. da Fonseca
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Correspondence to Carlos M. da Fonseca.

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Fonseca, C.M.d. On a closed form for derangement numbers: an elementary proof. RACSAM 114, 146 (2020). https://doi.org/10.1007/s13398-020-00879-3

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  • DOI: https://doi.org/10.1007/s13398-020-00879-3

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