Skip to main content
SpringerLink
Log in

Seven (Lattice) Paths to Log-Convexity

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Three new methods for proving log-convexity of combinatorial sequences are presented. Their implementation is demonstrated and their performance is compared with four more familiar approaches in the context of sequences that enumerate various classes of lattice paths.

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. Aigner, M.: Motzkin numbers. Eur. J. Comb. 19, 663–675 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  2. André, D.: Solution directe du problème résolu par M. Bertrand. C.R. Math. Acad. Sci. Paris 105, 436–437 (1887)

    Google Scholar 

  3. Angulo, J.C., Dehesa, J.S.: Atomic-charge log-convexity and radial expectation values. J. Phys. B 24, L299–L306 (1991)

    Article  Google Scholar 

  4. Artin, E.: The Gamma Function. Holt, Rinehart and Winston, New York (1964)

    MATH  Google Scholar 

  5. Asai, N., Kubo, I., Kuo, H.-H.: Bell numbers, log-concavity, and log-convexity. Acta Appl. Math. 63, 79–87 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Asai, N., Kubo, I., Kuo, H.-H.: Roles of log-concavity, log-convexity, and growth order in white noise analysis. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 4, 59–84 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Banderier, C., Flajolet, P.: Basic analytic combinatorics of directed lattice paths. Theor. Comput. Sci. 281, 37–80 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  8. Banderier, C., Schwer, S.: Why Delannoy numbers. J. Stat. Plan. Inference 135, 40–54 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Barcucci, E., Pinzani, R., Sprugnoli, R.: The Motzkin family. Pure Math. Appl. Ser. A 2, 249–279 (1991)

    MATH  MathSciNet  Google Scholar 

  10. Baryshnikov, Yu.: IT security investment and Gordon-Loeb’s 1/e rule. Preprint

  11. Bender, E.A., Canfield, E.R.: Log-concavity and related properties of the cycle index polynomials. J. Comb. Theory A 74, 56–70 (1996)

    MathSciNet  Google Scholar 

  12. Boche, H., Stanczak, S.: Log-convexity of the minimum total power in CDMA systems with certain quality-of-service guaranteed. IEEE Trans. Inf. Theory 51, 374–381 (2005)

    Article  MathSciNet  Google Scholar 

  13. Bohr, H., Mollerup, J.: Lærebog i Matematisk Analyse, vol. III. Jul. Gjellerups Forlag, Copenhagen (1922)

    Google Scholar 

  14. Bóna, M., Ehrenborg, R.: A combinatorial proof of the log-concavity of the numbers of permutations with k runs. J. Comb. Theory A 90, 293–303 (2000)

    Article  MATH  Google Scholar 

  15. Brenti, F.: Unimodal, Log-Concave and Pólya Frequency Sequences in Combinatorics. Am. Math. Soc., Providence (1989)

    Google Scholar 

  16. Callan, D.: Notes on Motzkin and Schröder numbers. Preprint (2000)

  17. Catalan, E.: Note sur une équation aux différences finies. J. Math. Pures Appl. 3, 508–516 (1838)

    Google Scholar 

  18. Catalan, E.: Sur les nombres de Segner. Rend. Circ. Mat. Palermo 1, 190–201 (1887)

    Article  Google Scholar 

  19. Deutsch, E.: Dyck path enumeration. Discrete Math. 204, 167–202 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  20. Došlić, T.: Log-balanced combinatorial sequences. Int. J. Math. Math. Sci. 2005(4), 507–522 (2005)

    Article  MATH  Google Scholar 

  21. Došlić, T., Veljan, D.: Calculus proofs of some combinatorial inequalities. Math. Inequal. Appl. 6, 197–210 (2003)

    MATH  MathSciNet  Google Scholar 

  22. Došlić, T., Veljan, D.: Logarithmic behavior of some combinatorial sequences. Discrete Math. 308, 2182–2212 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Došlić, T., Svrtan, D., Veljan, D.: Enumerative aspects of secondary structures. Discrete Math. 285, 67–82 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  24. Dulucq, S., Penaud, J.-G.: Interprétation bijective d’une récurrence des nombres de Motzkin. Discrete Math. 256, 671–676 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  25. Klauder, J.R., Penson, K.A., Sixdeniers, J.M.: Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems. Phys. Rev. A 64, 013817–013835 (2001)

    Article  Google Scholar 

  26. Koshy, T.: Catalan Numbers with Applications. Oxford University Press, Oxford (2008)

    Book  Google Scholar 

  27. Kuo, H.H.: White Noise Distribution Theory. Probability and Stochastic Series. CRC Press, Boca Raton (1996)

    MATH  Google Scholar 

  28. Kurtz, D.C.: A note on concavity properties of triangular arrays of numbers. J. Comb. Theory A 13, 135–159 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  29. Liu, L.L., Wang, Y.: On the log-convexity of combinatorial sequences. Adv. Appl. Math. 39, 453–476 (2007)

    Article  MATH  Google Scholar 

  30. Milenkovic, O., Compton, K.J.: Probabilistic transforms for combinatorial urn models. Comb. Probab. Comput. 13, 645–675 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. Mohanty, S.G.: Lattice Path Counting and Applications. Academic Press, San Diego (1979)

    MATH  Google Scholar 

  32. Motzkin, T.: Relation between hypersurface cross ratios, and a combinatorial formula for partitions of a polygon, for permanental preponderance and for non-associative products. Bull. Am. Math. Soc. 54, 352–360 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  33. Munarini, E., Zagaglia Salvi, N.: Binary strings without zigzags. In: Seminaire Lotharingien de Combinatoire, B49h (2004), 15 pp.

  34. Narayana, T.V.: Lattice Path Combinatorics with Statistical Applications. Toronto Univ. Press, Toronto (1979)

    MATH  Google Scholar 

  35. Netto, E.: Kombinatorik. In: Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen. Teubner, Leipzig (1898)

    Google Scholar 

  36. Penson, K.A., Solomon, A.I.: Coherent states from combinatorial sequences. In: Quantum Theory and Symmetries (Krakow 2001), pp. 527–530. World Scientific, Singapore (2002)

    Google Scholar 

  37. Prelec, D.: Decreasing impatience: a criterion for non-stationary time preference and ‘hyperbolic’ discounting. Scand. J. Econ. 106, 511–532 (2004)

    Article  Google Scholar 

  38. Riordan, J.: Percy Alexander MacMahon: collected papers, book review. Bull. Am. Math. Soc. New Ser. 2, 239–241 (1980)

    Article  MathSciNet  Google Scholar 

  39. Sagan, B.: Inductive and injective proofs of log-concavity results. Discrete Math. 68, 281–292 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  40. Schröder, E.: Vier kombinatorische Probleme. Z. Math. Phys. 15, 361–376 (1870)

    Google Scholar 

  41. Schuster, P., Stadler, P.F.: Discrete models of biopolymers. In: Crabbe, J., Konopka, A., Drew, M. (eds.) Handbook of Computational Chemistry and Biology, pp. 187–221. Dekker, New York (2004)

    Google Scholar 

  42. Schwer, S.R., Autebert, J.-M.: Henri-Auguste Delannoy, une biographie. Math. Soc. Sci. 43, 25–67 (2006)

    MathSciNet  Google Scholar 

  43. Shapiro, L.W., Sulanke, R.A.: Bijections for Schröder numbers. Math. Mag. 73, 369–376 (2000)

    Article  MathSciNet  Google Scholar 

  44. Sloane, N.J.A.: The on-line encyclopedia of integer sequences. Published electronically at http://www.research.att.com/~njas/sequences/index.html

  45. Stanley, R.P.: Log-concave and unimodal sequences in algebra, combinatorics and geometry. Ann. N.Y. Acad. Sci. 576, 500–535 (1989)

    Article  MathSciNet  Google Scholar 

  46. Stanley, R.P.: Hipparchus, Plutarch, Schröder, and Hough. Am. Math. Mon. 104, 344–350 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  47. Stanley, R.P.: Enumerative Combinatorics, vol. 2. Cambridge Univ. Press, Cambridge (1999)

    Google Scholar 

  48. Stanley, R.P.: Catalan addendum. Available at http://www-math.mit.edu/~rstan/ec/catadd.pdf

  49. Stein, P.R., Waterman, M.S.: On some new sequences generalizing the Catalan and Motzkin numbers. Discrete Math. 26, 261–272 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  50. Sulanke, R.A.: Objects counted by the central Delannoy numbers. J. Integer Seq. 3, 00.2.1 (2000)

    MathSciNet  Google Scholar 

  51. Szegö, G.: Orthogonal Polynomials. AMS, New York (1959)

    MATH  Google Scholar 

  52. Takacs, L.: On the ballot theorems. In: Balakrishnan, N. (ed.) Advances in Combinatorial Methods and Applications to Probability and Statistics, pp. 98–114. Birkhäuser, Boston (1979)

    Google Scholar 

  53. Warde, W.D., Katti, S.K.: Infinite divisibility of discrete distributions II. Ann. Math. Stat. 42, 1088–1090 (1964)

    Article  MathSciNet  Google Scholar 

  54. Weiss, M.: Theorems on log-convex disposition curves in drug and tracer kinetics. J. Theor. Biol. 116, 355–368 (1985)

    Article  Google Scholar 

  55. Weiss, M.: Generalizations in linear pharmacokinetics using properties of certain classes of residence time distributions. I. Log-convex drug disposition curves. J. Pharmacokinet. Biopharm. 14, 635–657 (1986)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Civil Engineering, University of Zagreb, Kačićeva 26, 10000, Zagreb, Croatia

    Tomislav Došlić

Authors
  1. Tomislav Došlić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tomislav Došlić.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Došlić, T. Seven (Lattice) Paths to Log-Convexity. Acta Appl Math 110, 1373–1392 (2010). https://doi.org/10.1007/s10440-009-9515-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-009-9515-4

Keywords

Mathematics Subject Classification (2000)

Search

Navigation