Glasnik Matematicki, Vol. 58, No. 1 (2023), 1-16. $$

INEQUALITIES ASSOCIATED WITH THE BAXTER NUMBERS

James Jing Yu Zhao

School of Accounting, Guangzhou College of Technology and Business, Foshan, Guangdong 528138, China
e-mail:zhao@gzgs.edu.cn

Abstract.   The Baxter numbers $B_n$ enumerate a lot of algebraic and combinatorial objects such as the bases for subalgebras of the Malvenuto-Reutenauer Hopf algebra and the pairs of twin binary trees on $n$ nodes. The Turán inequalities and higher order Turán inequalities are related to the Laguerre-Pólya ($\mathcal{L}$-$\mathcal{P}$) class of real entire functions, and the $\mathcal{L}$-$\mathcal{P}$ class has a close relation with the Riemann hypothesis. The Turán type inequalities have received much attention. In this paper, we are mainly concerned with Turán type inequalities, or more precisely, the log-behavior, and the higher order Turán inequalities associated with the Baxter numbers. We prove the Turán inequalities (or equivalently, the log-concavity) of the sequences $\{B_{n+1}/B_n{\}}_{n⩾0}$ and $\{\sqrt[n]{B_n}{\}}_{n⩾1}$. Monotonicity of the sequence $\{\sqrt[n]{B_n}{\}}_{n⩾1}$ is also obtained. Finally, we prove that the sequences $\{B_n/n!{\}}_{n⩾2}$ and $\{B_{n+1}{B}_n^{-1}/n!{\}}_{n⩾2}$ satisfy the higher order Turán inequalities.

2020 Mathematics Subject Classification.   05A20, 11B83

Key words and phrases.   Log-concavity, log-convexity, log-balancedness, higher order Turán inequalities, Baxter numbers

https://doi.org/10.3336/gm.58.1.01

References:

1. J.-C. Aval, A. Boussicault, M. Bouvel, O. Guibert and M. Silimbani, Baxter tree-like tableaux, arXiv.2108.06212v1, 2021.

2. G. Baxter, On fixed points of the composite of commuting functions, Proc. Amer. Math. Soc. 15 (1964), 851–855.
   MathSciNet    CrossRef

3. G. D. Birkhoff and W.J. Trjitzinsky, Analytic theory of singular difference equations, Acta Math. 60 (1933), 1–89.
   MathSciNet    CrossRef

4. R. P. Boas Jr., Entire functions, Academic Press Inc., New York, 1954.
   MathSciNet

5. G. Chatel and V. Pilaud, Cambrian Hopf algebras, Adv. Math. 311 (2017), 598–633.
   MathSciNet    CrossRef

6. W. Y. C. Chen, The spt-function of Andrews, in: Surveys in Combinatorics 2017, eds. Claesson, A., Dukes, M., Kitaev, S., Manlove, D., Meeks, K., London Math. Soc. Lecture Note Ser. 440, Cambridge Univ. Press, Cambridge, 2017, 141–203.
   MathSciNet

7. W. Y. C. Chen, J. J. F. Guo and L. X. W. Wang, Infinitely log-monotonic combinatorial sequences, Adv. in Appl. Math. 52 (2014), 99–120.
   MathSciNet    CrossRef

8. W. Y. C. Chen, D. X. Q. Jia and L. X. W. Wang, Higher order Turán inequalities for the partition function, Trans. Amer. Math. Soc. 372 (2019), 2143–2165.
   MathSciNet    CrossRef

9. F. R. K. Chung, R. L. Graham, V. E. Hoggatt Jr. and M. Kleiman, The number of Baxter permutations, J. Combin. Theory Ser. A 24 (1978), 382–394.
   MathSciNet    CrossRef

10. R. Cori, S. Dulucq and G. Viennot, Shuffle of Parenthesis Systems and Baxter Permutations, J. Combin. Theory Ser. A 43 (1986), 1–22.
    MathSciNet    CrossRef

11. J. Courtiel, E. Fusy, M. Lepoutre and M. Mishna, Bijections for Weyl Chamber walks ending on an axis, using arc diagrams and Schnyder woods, European J. Combin. 69 (2018), 126–142.
    MathSciNet    CrossRef

12. T. Craven and G. Csordas, Jensen polynomials and the Turán and Laguerre inequalities, Pacific J. Math. 136 (1989), 241–260.
    MathSciNet    Link

13. G. Csordas, T.S. Norfolk and R.S. Varga, The Riemann hypothesis and the Turán inequalities, Trans. Amer. Math. Soc. 296 (1986), 521–541.
    MathSciNet    CrossRef

14. G. Csordas and R.S. Varga, Necessary and sufficient conditions and the Riemann hypothesis, Adv. in Appl. Math. 11 (1990), 328–357.
    MathSciNet    CrossRef

15. D.K. Dimitrov, Higher order Turán inequalities, Proc. Amer. Math. Soc. 126 (1998), 2033–2037.
    MathSciNet    CrossRef

16. D. K. Dimitrov and F.R. Lucas, Higher order Turán inequalities for the Riemann $\xi$-function, Proc. Amer. Math. Soc. 139 (2011), 1013–1022.
    MathSciNet    CrossRef

17. T. Došlić, Log-balanced combinatorial sequences, Intl. J. Math. Math. Sci. 4 (2005), 507–522.
    MathSciNet    CrossRef

18. T. Došlić, D. Svrtan and D. Veljan, Enumerative aspects of secondary structures, Discrete Math. 285 (2004), 67–82.
    MathSciNet    CrossRef

19. T. Došlić and D. Veljan, Calculus proofs of some combinatorial inequalities, Math. Inequal. Appl. 6 (2003), 197–209.
    MathSciNet    CrossRef

20. T. Došlić and D. Veljan, Logarithmic behavior of some combinatorial sequences, Discrete Math. 308 (2008), 2182–2212.
    MathSciNet    CrossRef

21. S. Dulucq and O. Guibert, Baxter permutations, Discrete Math. 180 (1998), 143–156.
    MathSciNet    CrossRef

22. S. Felsner, É. Fusy, M. Noy and D. Orden, Bijections for Baxter families and related objects, J. Combin. Theory Ser. A 118 (2011), 993–1020.
    MathSciNet    CrossRef

23. S. Giraudo, Algebraic and combinatorial structures on pairs of twin binary trees, J. Algebra 360 (2012), 115–157.
    MathSciNet    CrossRef

24. Q.-H. Hou and Z.-R. Zhang, Asymptotic $r$-log-convexity and $P$-recursive sequences, J. Symbolic Comput. 93 (2019), 21–33.
    MathSciNet    CrossRef

25. Q.-H. Hou and G. Li, Log-concavity of $P$-recursive sequences, J. Symbolic Comput. 107 (2021), 251–268.
    MathSciNet    CrossRef

26. S. Law and N. Reading, The Hopf algebra of diagonal rectangulations, J. Combin. Theory Ser. A 119 (2012), 788–824.
    MathSciNet    CrossRef

27. L. L. Liu and Y. Wang, On the log-convexity of combinatorial sequences, Adv. in Appl. Math. 39 (2007), 453–476.
    MathSciNet    CrossRef

28. G. V. Milovanović, D. S. Mitrinović and Th. M. Rassias, Topics in polynomials: extremal problems, inequalities, zeros, World Scientific Publishing Co., Inc., River Edge, NJ, 1994.
    MathSciNet    CrossRef

29. C. P. Niculescu, A new look at Newton's inequalities, J. Inequal. Pure Appl. Math. 1 (2000), Article 17, 14.
    MathSciNet

30. G. Pólya, Über die algebraisch-funktionentheoretischcen Untersuchungen von J. L. W. V. Jensen, Kgl. Danske Vid. Sel. Math.-Fys. Medd. 7 (1927), 3–33.

31. N. Reading, Lattice congruences, fans and Hopf algebras, J. Combin. Theory Ser. A 110 (2005), 237–273.
    MathSciNet    CrossRef

32. P. Ribenboim, The Little Book of Bigger Primes, Second Edition, Springer-Verlag, New York, 2004.
    MathSciNet

33. S. Rosset, Normalized symmetric functions, Newton's inequalities and a new set of stronger inequalities, Amer. Math. Monthly 96 (1989), 815–819.
    MathSciNet    CrossRef

34. W. Rudin, Principles of mathematical analysis, third edition, McGraw-Hill Book Co., New York-Auckland-DĂĽsseldorf, 1976.
    MathSciNet

35. N. J. A. Sloane, The On-line encyclopedia of integer sequences, .

36. Z.-W. Sun, Conjectures involving arithmetical sequences, in: Number theory–arithmetic in Shangri-La,
    Link    MathSciNet    CrossRef

37. Z.-W. Sun, New conjectures in number theory and combinatorics, Harbin institute of Technology Press, 2021.

38. G. Szegö, On an inequality of P. Turán concerning Legendre polynomials, Bull. Amer. Math. Soc. 54 (1948), 401–405.
    MathSciNet    CrossRef

39. G. Viennot, A bijective proof for the number of Baxter permutations, in: Troisième Séminaire Lotharingien de Combinatoire, Le Klebach 1981, 28–29.

40. L. X. W. Wang, Higher order Turán inequalities for combinatorial sequences, Adv. in Appl. Math. 110 (2019), 180–196.
    MathSciNet    CrossRef

41. J. Wimp and D. Zeilberger, Resurrecting the asymptotics of linear recurrences, J. Math. Anal. Appl. 111 (1985), 162–176.
    MathSciNet    CrossRef

42. D. Zeilberger, The method of creative telescoping, J. Symbolic Comput. 11 (1991), 195–204.
    MathSciNet    CrossRef