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Arboretum for a generalisation of Ramanujan polynomials

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Abstract

In this paper, we expand on the work of Guo and Zeng (Adv Appl Math 39(1):96–115, 2007) on a generalisation of the Ramanujan polynomials and planar trees. We manage to find combinatorial interpretations of this family of polynomials in terms of Greg trees, Cayley trees and planar trees by constructing bijections that preserve relevant tree statistics.

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References

  1. Berndt, B.C., Evans, R.J., Wilson, B.M.: Chapter 3 of Ramanujan’s second notebook. Adv. Math. 49(2), 123–169 (1983). https://doi.org/10.1016/0001-8708(83)90071-3

    Article  MathSciNet  MATH  Google Scholar 

  2. Berndt, B.C.: Ramanujan’s Second Notebooks, Part I, Chap. 3. Combinatorial Analysis and Series Inversions. Springer, New York (1985)

    MATH  Google Scholar 

  3. Billera, L.J., Holmes, S.P., Vogtmann, K.: Geometry of the space of phylogenetic trees. Adv. Appl. Math. 27(4), 733–767 (2001)

    Article  MathSciNet  Google Scholar 

  4. Chapoton, F.: Opérades différentielles graduées sur les simplexes et les permutoedres. Bull. Soc. Math. Fr. 130(2), 233–252 (2002)

    Article  Google Scholar 

  5. Chen, W.Y., Guo, V.J.: Bijections behind the Ramanujan polynomials. arXiv:math/0107024 (2001)

  6. Dumont, D., Ramamonjisoa, A.: Grammaire de Ramanujan et arbres de Cayley. Electron. J. Comb. 3, 2 (1996)

    MathSciNet  MATH  Google Scholar 

  7. Flight, C.: How many stemmata? Manuscripta 34(2), 122–128 (1990)

    Article  Google Scholar 

  8. Guo, V.J., Zeng, J.: A generalization of the Ramanujan polynomials and plane trees. Adv. Appl. Math. 39(1), 96–115 (2007)

    Article  MathSciNet  Google Scholar 

  9. Josuat-Vergès, M.: Derivatives of the tree function. Ramanujan J. 38(1), 1–15 (2015)

    Article  MathSciNet  Google Scholar 

  10. Ramanujan, S., et al.: Notebooks of Srinivasa Ramanujan, vol. 2. Springer, New York (2013)

    MATH  Google Scholar 

  11. Reddy, U.S.: Term rewriting induction. In: International Conference on Automated Deduction, pp. 162–177. Springer, New York (1990)

  12. Shor, P.W.: A new proof of Cayley’s formula for counting labeled trees. J. Comb. Theory Ser. A 71(1), 154–158 (1995)

    Article  MathSciNet  Google Scholar 

  13. Speyer, D., Sturmfels, B.: The tropical grassmannian. Adv. Geom. 4(3), 389–411 (2004)

    Article  MathSciNet  Google Scholar 

  14. Zeng, J.: A Ramanujan sequence that refines the Cayley formula for trees. Ramanujan J. 3(1), 45–54 (1999)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank Matthieu Josuat-Vergès for his invaluable help and support.

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  1. Université Paris-Est Marne-la-Vallée, Bâtiment UPEM Copernic 4B03R, Champs-sur-Marne, France

    Lucas Randazzo

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  1. Lucas Randazzo
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Correspondence to Lucas Randazzo.

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Randazzo, L. Arboretum for a generalisation of Ramanujan polynomials. Ramanujan J 54, 591–604 (2021). https://doi.org/10.1007/s11139-019-00185-6

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  • DOI: https://doi.org/10.1007/s11139-019-00185-6

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