Skip to main content
SpringerLink
Log in

Recurrences for Callan’s Generalization of Narayana Polynomials

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

By using Chen, Hou and Mu’s extended Zeilberger algorithm, the authors obtain two recurrence relations for Callan’s generalization of Narayana polynomials. Based on these recurrence relations, the authors further prove the real-rootedness and asymptotic normality of Callan’s Narayana polynomials.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Sloane N J A, The on-line encyclopedia of integer sequences, 1964, https://oeis.org.

  2. Catalan E, Sur les Nombres de Segner, Rend. Circ. Mat. Palermo, 1887, 1: 190–201.

    Article  Google Scholar 

  3. Stanley R P, Enumerative Combinatorics, Vol. 1, 2nd Ed., Cambridge University Press, Cambridge, 2012.

    MATH  Google Scholar 

  4. Bóna M and Sagan B E, On divisibility of Narayana numbers by primes, J. Integer Seq., 2005, 8: Article 05.2.4.

  5. MacMahon P A, Combinatorial Analysis, Vols. 1 and 2, Cambridge University Press, 1915, 1916; reprinted by Chelsea, 1960.

  6. Narayana T V, Sur les treillis formés par les partitions d’une unties et leurs applications à la théorie des probabilités, C. R. Acad. Sci. Paris, 1955, 240: 1188–1189.

    MathSciNet  MATH  Google Scholar 

  7. Armstrong D, Generalized noncrossing partitions and combinatorics of coxeter groups, Mem. Amer. Math. Soc., 2009, 202(949): x+159.

    MathSciNet  MATH  Google Scholar 

  8. Chen H Z Q, Yang A L B, and Zhang P B, Kirillov’s unimodality conjecture for the rectangular Narayana polynomials, Electron. J. Combin., 2018, 25(1): P1.17.

    MathSciNet  MATH  Google Scholar 

  9. Chen H Z Q, Yang A L B, and Zhang P B, The real-rootedness of generalized Narayana polynomials related to the Boros-Moll polynomials, Rocky Mountain J. Math., 2018, 48(1): 107–119.

    Article  MathSciNet  Google Scholar 

  10. Narayana T V, Lattice Path Combinatorics with Statistical Applications, Mathematical Expositions 23., University of Toronto Press, Toronto, 1979.

    MATH  Google Scholar 

  11. Sulanke R A, The Narayana distribution, J. Statist. Plann. Inference, 2002, 101: 311–326.

    Article  MathSciNet  Google Scholar 

  12. Sulanke R A, Generalizing Narayana and Schröder numbers to higher dimensions, Electron. J. Combin., 2004, 11: #R54.

    Article  Google Scholar 

  13. Callan D, Generalized Narayana numbers, preprint, https://www.oeis.org/A281260/a281260.pdf, 2017.

  14. Chen X, Liang H, and Wang Y, Total positivity of recursive matrices, Linear Algebra Appl., 2015, 471: 383–393.

    Article  MathSciNet  Google Scholar 

  15. Wang Y and Yang A L B, Total positivity of Narayana matrices, Discrete Math., 2018, 341(5): 1264–1269.

    Article  MathSciNet  Google Scholar 

  16. Liu L L and Wang Y, A unified approach to polynomoal sequences with only real zeros, Adv. Appl. Math., 2007, 38: 542–560.

    Article  Google Scholar 

  17. Chen X, Mao J, and Wang Y, Asymptotic normality in t-stack sortable permutations, Proc. Edinb. Math. Soc. (2), 2020, 63(4): 1062–1070.

    Article  MathSciNet  Google Scholar 

  18. Chen W Y C, Hou Q H, and Mu Y P, The extended Zeilberger algorithm with parameters, J. Symbolic Comput., 2012, 47(6): 643–654.

    Article  MathSciNet  Google Scholar 

  19. Bender E A, Central and local limit theorems applied to asymptotic enumeration, J. Combin. Theory Ser. A, 1973, 15: 91–111.

    Article  MathSciNet  Google Scholar 

  20. Harper L H, Stirling behavior is asymptotically normal, Ann. Math. Statist., 1967, 38: 410–414.

    Article  MathSciNet  Google Scholar 

  21. Shapiro L, Some open questions about random walks, involutions, limiting distributions, and generating functions, Adv. Appl. Math., 2001, 27: 585–596.

    Article  MathSciNet  Google Scholar 

  22. Chen S and Kauers M, Some open problems related to creative telescoping, Journal of Systems Science and Complexity, 2017, 30(1): 154–172.

    Article  MathSciNet  Google Scholar 

  23. Hou Q H, Maple package APCI, http://faculty.tju.edu.cn/HouQinghu/en/index.htm, 2012.

  24. Koutschan C, Advanced applications of the holonomic systems approach, Doctoral Thesis, Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Linz, Austria, 2009.

    MATH  Google Scholar 

  25. Koutschan C, The Mathematica package “HolonomicFunctions.m”, https://www3.risc.jku.at/research/combinat/software/ergosum/RISC/HolonomicFunctions.html, 2009.

  26. Sulanke R A, Three recurrences for parallelogram polyominoes, J. Diff. Equations Appl., 1999, 5: 155–176.

    Article  MathSciNet  Google Scholar 

  27. Brändén P, The generating function of two-stack sortable permutations by descents is real-rooted, arXiv: math/0303149v2, 2003.

  28. Gasper G and Rahman M, Basic Hypergeometric Series, Second Edition, Cambridge University Press, Cambridge, 2004.

    Book  Google Scholar 

  29. Rainville E D, Special Functions, The Macmillan Company, New York, 1960.

    MATH  Google Scholar 

  30. Canfield E R, Central and local limit theorems for the coefficients of polynomials of binomial type, J. Combin. Theory Ser. A, 1977, 23: 275–290.

    Article  MathSciNet  Google Scholar 

  31. Gould H W, Some generalizations of Vandermonde’s convolution, Amer. Math. Monthly, 1956, 63(2): 84–91.

    Article  MathSciNet  Google Scholar 

  32. Graham R L, Knuth D E, and Patashnik O, Concrete Mathematics: A Foundation for Computer Science, 2nd Edition, Addison-Wesley Professional, New York, 1994.

    MATH  Google Scholar 

  33. Spencer J and Florescu L, Asymptopia, Student mathematical library Vol. 71., American Mathematical Society, Providence, Rhode Island, 2014.

    Google Scholar 

Download references

Acknowledgements

We are grateful to the anonymous referees for their helpful comments and suggestions. We also would like to thank Qing-Hu Hou for his valuable comments, especially for showing us the extended Zeilberger algorithm is applicable in finding recurrence relations for certain combinatorial sequences.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China

    Xi Chen

  2. Center for Combinatorics, LPMC, Nankai University, Tianjin, 300071, China

    Arthur Li Bo Yang

  3. School of Mathematics, Tianjin University, Tianjin, 300350, China

    James Jing Yu Zhao

Authors
  1. Xi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Arthur Li Bo Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. James Jing Yu Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James Jing Yu Zhao.

Additional information

CHEN was supported by the National Natural Science Foundation of China under Grant No. 11601062. YANG was supported in part by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China under Grant Nos. 11522110 and 11971249, respectively. ZHAO was partially supported by the National Natural Science Foundation of China under Grant Nos. 11771330 and 11971203.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, X., Yang, A.L.B. & Zhao, J.J.Y. Recurrences for Callan’s Generalization of Narayana Polynomials. J Syst Sci Complex 35, 1573–1585 (2022). https://doi.org/10.1007/s11424-021-0216-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-021-0216-z

Keywords

Search

Navigation