The area determined by underdiagonal lattice paths

  • Donatella Merlini
  • Renzo Sprugnoli
  • M. Cecilia Verri
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1059)


We use the “first passage decomposition” methodology to study the area between various kinds of underdiagonal lattice paths and the main diagonal. This area is important because it is connected to the number of inversions in permutations and to the internal path length in various types of trees. We obtain the generating functions for the total area of all the lattice paths from the origin to the point (n, n). Since this method also determines the number of these paths, we are able to obtain exact results for the average area.


underdiagonal lattice paths average area first passage decomposition generating functions context-free grammars 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    E. Barcucci, R. Pinzani, and R. Sprugnoli. The Motzkin family. Pure Mathematics and Applications, 2:249–279, 1991.Google Scholar
  2. 2.
    L. Carlitz. Sequences, paths, ballot numbers. Fibonacci Quart., 10:531–549, 1972.Google Scholar
  3. 3.
    L. Carlitz and J. Riordan. Two elements lattice permutation numbers and their q-generalization. Duke J. Math., 31:371–388, 1964.Google Scholar
  4. 4.
    M. P. Delest and J. M. Fédou. Enumeration of skew Ferrers diagrams. Discrete Mathematics, 112:65–79, 1993.Google Scholar
  5. 5.
    M. P. Delest and G. Viennot. Algebraic languages and polyominoes. Theoretical Computer Science, 34:169–206, 1984.Google Scholar
  6. 6.
    I. Dutour and J. M. Fédou. Grammaire d'objects. Technical Report 963, LaBRI, Université Bordeaux I, 1994.Google Scholar
  7. 7.
    W. Feller. An introduction to probability theory and its applications. Wiley, 1950.Google Scholar
  8. 8.
    J. Francon and X. G. Viennot. Permutation selon les pics, creux, doubles montees, doubles descentes, nombres d'Euler et nombres de Genocchi. Discrete Mathematics, 28:21–35, 1979.Google Scholar
  9. 9.
    R. D. Fray and D. P. Roselle. Weighted lattice paths. Pacific Journal of Mathematics, 37:85–96, 1971.Google Scholar
  10. 10.
    R. D. Fray and D. P. Roselle. On weighted lattice paths. Journal of Combinatorial Theory, series A, 14:21–29, 1973.Google Scholar
  11. 11.
    J. Fürlinger and J. Holfbauer. q-Catalan numbers. Journal of Combinatorial Theory, Series A, 40:248–264, 1985.Google Scholar
  12. 12.
    J. R. Goldman. Formal languages and enumeration. Journal of Combinatorial Theory, Series A, 24:318–338, 1978.Google Scholar
  13. 13.
    J. R. Goldman and T. Sundquist. Lattice path enumeration by formal schema. Advances in applied mathematics, 13:216–251, 1992.Google Scholar
  14. 14.
    E. Goodman and T. V. Narayana. Lattice paths with diagonal steps. Canad. Math. Bull., 12:847–855, 1969.Google Scholar
  15. 15.
    I. P. Goulden and D. M. Jackson. Combinatorial Enumeration. John Wiley & S., 1983.Google Scholar
  16. 16.
    B. R. Handa and S. G. Mohanty. Higher dimensional lattice paths with diagonal steps. Discrete Mathematics, 15:137–140, 1976.Google Scholar
  17. 17.
    D. E. Knuth. The art of computer programming. Vol. 1–3. Addison-Wesley, 1973.Google Scholar
  18. 18.
    J. Labelle and Y. Yeh. Dyck paths of knight moves. Discrete applied mathematics, 24:213–221, 1989.Google Scholar
  19. 19.
    J. Labelle and Y. Yeh. Generalized Dyck paths. Discrete mathematics, 82:1–6, 1990.Google Scholar
  20. 20.
    D. Merlini, D. G. Rogers, R. Sprugnoli, and M. C. Verri. Lattice paths with steep and shallow steps. Technical Report 16, Dipartimento di Sistemi e Informatica, Università di Firenze, 1995.Google Scholar
  21. 21.
    D. Merlini, R. Sprugnoli, and M. C. Verri. Algebraic and combinatorial properties of simple, coloured walks. In Proceedings of CAAP'94, volume 787 of Lecture Notes in Computer Science, pages 218–233, 1994.Google Scholar
  22. 22.
    S. G. Mohanty and B. R. Handa. On lattice paths with several diagonal steps. Canad. Math. Bull., 11:537–545, 1968.Google Scholar
  23. 23.
    L. Moser and W. Zayachkowski. Lattice paths with diagonal steps. Scripta Math., 26:223–229, 1963.Google Scholar
  24. 24.
    D. G. Rogers and L. W. Shapiro. Deques, trees and lattice paths. Lectures Notes in Mathematics, 884:292–303, 1981.Google Scholar
  25. 25.
    V. K. Rohatgi. On lattice paths with diagonals steps. Canad. Math. Bull., 7:470–472, 1964.Google Scholar
  26. 26.
    M. P. Schützenberger. Context-free language and pushdown automata. Information and Control, 6:246–264, 1963.Google Scholar
  27. 27.
    L. Takács. Some asymptotic formulas for lattice paths. Journal of statistical planning and inference, 14:123–142, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Donatella Merlini
    • 1
  • Renzo Sprugnoli
    • 1
  • M. Cecilia Verri
    • 1
  1. 1.Dipartimento di Sistemi e InformaticaFirenzeItaly

Personalised recommendations