Enumerative combinatorics on words

  • D. Perrin


Generating series, also called generating functions, play an important role in combinatorial mathematics. Many enumeration problems can be solved by transferring the basic operations on sets into algebraic operations on formal series leading to a solution of an enumeration problem. The famous paper by Doubilet, Rota and Stanley, The idea of generating function [40], places the subject in a general mathematical frame-work allowing one to present in a unified way the diversity of generating functions, from the ordinary ones to the exponential or even Dirichlet.


Zeta Function Adjacency Matrix Length Distribution Finite Type Finite Automaton 
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  1. [1]
    Adler, R.L., Coppersmith, D., Hassner, M. (1983): Algorithms for sliding block codes. An application of symbolic dynamics to information theory. IEEE Trans. Inform. Theory 29, 5–22MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Ahlswede, R., Balkenhol, B., Khachatrian, L. (1997): Some properties of fix-free codes. Preprint 97-39. Fakultät für Mathematik, Universität BielefeldGoogle Scholar
  3. [3]
    Aigner, M., Ziegler, G.M. (1998): Proofs from The Book. Springer, BerlinMATHGoogle Scholar
  4. [4]
    Ash, R.B. (1990): Information theory. Dover, New YorkMATHGoogle Scholar
  5. [5]
    Ashley, J.J. (1998): A linear bound for sliding-block decoder window size. IEEE Trans. Inform. Theory 34, 389–399MathSciNetCrossRefGoogle Scholar
  6. [6]
    Bassino, F. (1999): Generating functions of circular codes. Adv. in Appl. Math. 22, 1–24MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Bassino, F., Béal, M.-P., Perrin, D. (1997): Enumerative sequences of leaves in rational trees. In: Degano, P. et al. (eds.) Automata, Languages and Programming. (Lecture Notes in Computer Science, vol. 1256), Springer, Berlin, pp. 76–86CrossRefGoogle Scholar
  8. [8]
    Bassino, F., Béal, M.-P., Perrin, D. (1999): Enumerative sequences of leaves and nodes in rational trees. Theoret. Comput. Sci. 221, 41–60MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Bassino, F., Béal, M.-P., Perrin, D. (1999): Length distributions and regular sequences, In: Rosenthal, J., Marcus, B. (eds.) Codes, Systems and Graphical Models. (IMA Volumes in Mathematics and its Applications, vol.123). Springer, New YorkGoogle Scholar
  10. [10]
    Bassino, F., Béal, M.-P., Perrin, D. (2000): A finite state version of the Kraft-McMillan theorem. SIAM J. Comput. 30, 1211–1230MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Béal, M.-P. (1993): Codage symbolique. Masson, ParisGoogle Scholar
  12. [12]
    Béal, M.-P. (1995): Puissance extérieure d’un automate déterministe, application au calcul de la fonction zêta d’un système sofique. RAIRO Inform. Théor. Appl. 29, 85–103MATHGoogle Scholar
  13. [13]
    Béal, M.-P., Mignosi, F., Restivo, A. (1996): Minimal forbidden words and symbolic dynamics. In: Puech, C., Reischuk, R. (eds.) STACS96. (Lecture Notes in Computer Science, vol. 1046). Springer, Berlin, pp. 555–566Google Scholar
  14. [14]
    Béal, M.-P., Mignosi, F., Restivo, A., Sciortino, M. (2000): Forbidden words in symbolic dynamics. Adv. in Appl. Math. 25, 163–193MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Béal, M.-P., Perrin, D. (1997): Symbolic dynamics and finite automata. In: Rosenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 2. Springer, Berlin, pp. 463–505Google Scholar
  16. [16]
    Berstel, J., Perrin, D. (1985): Theory of codes. Academic Press, Orlando, FLMATHGoogle Scholar
  17. [17]
    Berstel, J., Reutenauer, C. (1990): Zeta functions of formal languages. Trans. Amer. Math. Soc. 321, 533–546MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Berstel, J., Reutenauer, C. (1988): Rational series and their languages. Springer, BerlinMATHCrossRefGoogle Scholar
  19. [19]
    Bowen, R., Lanford, O.E. (1970): Zeta functions of restrictions of the shift transformation. In: Chern, S.-S., Smale, S. (eds.) Global Analysis. (Proceedings of Symposia in Pure Mathematics, vol. 14). American Mathematical Society, Providence, RI, pp. 43–49Google Scholar
  20. [20]
    Bowen, R. (1978): On Axiom A diffeomorphisms. (Regional Conferenc Series in Mathematics, no. 35). American Mathematical Society, Providence, RIMATHGoogle Scholar
  21. [21]
    Bruyère, V., Latteux, M. (1996): Variable-length maximal codes. In: Meyer auf der Heide, F., Monien, B. (eds.) Automata, Languages and Programming. (Lecture Notes in Computer Science, vol. 1099). Springer, Berlin, pp. 24–47CrossRefGoogle Scholar
  22. [22]
    Eilenberg, S. (1974): Automata, languages and machines, vol. A. Academic Press, New YorkMATHGoogle Scholar
  23. [23]
    Flajolet, P. (1987): Analytic models and ambiguity of context-free languages. Theoret. Comput. Sci. 49, 283–309MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Forney, G.D., Marcus, B.H., Sindhushayana, N.T., Trott, M. (1995): Multilingual dictionary: system theory, coding theory, symbolic dynamics, and automata theory. In: Calderbank, R. (ed.) Different Aspects of Coding Theory. (Proceedings of Symposia in Applied Mathematics, vol. 50). American Mathematical Society, Providence, RI, pp. 109–138Google Scholar
  25. [25]
    Gantmatcher, F.R. (1959): Theory of matrices, vol. 2. Chelsea, New YorkGoogle Scholar
  26. [26]
    Graham, R.L., Knuth, D., Patachnik, O. (1989): Concrete mathematics. A foundation for computer science. Addison Wesley, Reading, MAGoogle Scholar
  27. [27]
    Katayama, T., Okamoto, M., Enomoto, H. (1978): Characterization of the structuregenerating functions of regular sets and the DOL growth functions. Information and Control 36, 85–101MathSciNetMATHCrossRefGoogle Scholar
  28. [28]
    Kitchens, B. P. (1998): Symbolic dynamics. One-sided, two-sided and countable state Markov shifts. Springer, BerlinMATHGoogle Scholar
  29. [29]
    Lang, S. (1984): Algebra, 2nd edition. Addison Wesley, Reading, MAMATHGoogle Scholar
  30. [30]
    Lind, D.A., Marcus, B.H. (1995): An introduction to symbolic dynamics and coding. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  31. [31]
    Lothaire, M. (1997): Combinatorics on words. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  32. [32]
    Macdonald, I.G. (1995): Symmetric functions and Hall polynomials, 2nd edition. Clarendon Press, OxfordMATHGoogle Scholar
  33. [33]
    Manning, A. (1971): Axiom A diffeomorphisms have rational zeta functions. Bull. London Math. Soc. 3, 215–220MathSciNetMATHCrossRefGoogle Scholar
  34. [34]
    Marcus, B.H. (1979): Factors and extensions of full shifts. Monatsh. Math. 88, 239–247MathSciNetMATHCrossRefGoogle Scholar
  35. [35]
    Marcus, B.H., Roth, R.M., Siegel, PH. (1998): Constrained systems and coding for recording channels. In: Pless, V.S. et al. (eds.) Handbook of Coding Theory, vol. II. North Holland, Amsterdam, pp. 1635–1764Google Scholar
  36. [36]
    Metropolis, N., Rota, G.-C. (1983): Witt vectors and the algebra of necklaces. Adv. Math. 50, 95–125MathSciNetMATHCrossRefGoogle Scholar
  37. [37]
    Perrin, D. (1990): Finite Automata. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B. Elsevier, Amsterdam, pp. 1–57Google Scholar
  38. [38]
    Reutenauer, C. (1997): Personal communicationGoogle Scholar
  39. [39]
    Reutenauer, C. (1997): N-rationality of zeta functions. Adv. in Appl. Math. 29, 1–17MathSciNetCrossRefGoogle Scholar
  40. [40]
    Rota, G.-C. (1975): Finite operator calculus. Academic Press, New YorkMATHGoogle Scholar
  41. [41]
    Salomaa, A., Soittola, M. (1978): Automata-theoretic properties of formal power series. Springer, New YorkCrossRefGoogle Scholar
  42. [42]
    Scharf, T., Thibon, J.-Y. (1996): On Witt vectors and symmetric functions. Algebra Colloq. 3, 231–238MathSciNetMATHGoogle Scholar
  43. [43]
    Stanley, R.P. (1997): Enumerative combinatorics, vol. 1. Cambridge University Press, CambridgeMATHCrossRefGoogle Scholar
  44. [44]
    Ye, C., Yeung, R.W. (2000): Basic properties of fix-free codes. IEEE Trans. Inform. Theory, submittedGoogle Scholar

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  • D. Perrin

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