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Abstract

I met Gian-Carlo in 1976, landing in MIT trying to establish Western connections. I was already working with Marcel-Paul Schützenberger who was one of his great friends and I needed no further introduction. Moreover, the few dollars that I had in my pocket were forcing me to eat fast, and this was contrary to Gian-Carlo’s sense of hospitality. The outcome was that I appeared several times on his list of professional expenses (section: restaurants). It was still an epoch in which tax inspectors readily accepted supporting combinatorics. This is no longer the case since combinatorics received the imprimatur of the Bourbaki Seminar. Let me however point out that meals were followed by long discussions about the comparative merits of algebraic structures, Gian Carlo for his part tirelessly asking me to repeat the definition of λrings that he copied each time in his black notebook with a new illustrative example.

Keywords

Symmetric Function Divided Difference Bernoulli Number Cauchy Kernel Jack Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    ACE (Veigneau, S. et al.) (1998): An algebraic environment for the computer algebra system MAPLE; http://phalanstere.univ-mlv.fr (1998)
  2. [2]
    Brenti, F. (1999): A class of q-symmetric functions arising from plethysm. PreprintGoogle Scholar
  3. [3]
    Gessel, I., Viennot, X. (1985): Binomial determinants, paths, and hook length formulae. Adv. Math. 58, 300–321MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Knutson, D. (1973): λ-rings and the representation theory of the symmetric group. (Lecture Notes in Mathematics, vol. 308). Springer, BerlinMATHGoogle Scholar
  5. [5]
    Lascoux, A., Lassalle, M. (2000): Une identité remarquable en théorie des partitions. Math. Ann. To appearGoogle Scholar
  6. [6]
    Lascoux, A., Schützenberger, M.P. (1982): Polynômes de Schubert. C. R. Acad. Sci. Paris Sér. I Math. 294, 447–450MATHGoogle Scholar
  7. [7]
    Lascoux, A., Schützenberger, M.P. (1985): Formulaire raisonné de fonctions symétriques. Université de Paris VII, ParisGoogle Scholar
  8. [8]
    Macdonald, I.G. (1995): Symmetric functions and Hall polynomials, 2nd edition. Clarendon Press, OxfordMATHGoogle Scholar
  9. [9]
    Prosper, V. (1999): Combinatoire des polynômes multivariés, Thèse, Université de Marne-la-Vallée, Marne-la-Vallée; http://schubert.univ-mlv.fr
  10. [10]
    Prosper, V. (2000): SFA, a package on symmetric functions considered as operators over the ring of polynomials for the computer algebra system MAPLE. J. Symbolic Comput. 29, 83–94MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Riordan, J. (1968): Combinatorial identities. Wiley, New YorkMATHGoogle Scholar
  12. [12]
    Roman, S. (1985): More on the umbral calculus, with emphasis on the q-umbral calculus. J. Math. Anal. Appl. 107, 222–254MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Rota, G.-C., Kahaner, D., Odlyzko, A. (1973): Finite operator calculus. J. Math. Anal. Appl. 42, 684–760MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Rota, G.-C. (1964): On the foundations of combinatorial theory. I. Theory of Möbius functions. Z. Wahrsch. Verw. Gebiete 2, 340–368MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Rota, G.-C. (1964): The number of partitions of a set. Amer. Math. Monthly 71, 498–504MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Rota, G.-C. (1969): Baxter algebras and combinatorial identities. I, II. Bull. Amer. Math. Soc. 75, 325–329, 330-334MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Mullin, R., Rota, G.-C. (1970): On the foundations of combinatorial theory. III. Theory of binomial enumeration. In: Harris, B. (ed.) Graph Theory and its Applications. Academic Press, New York, pp. 167–213Google Scholar
  18. [18]
    Goldman, J., Rota, G.-C. (1970): On the foundations of combinatorial theory. IV. Finite vector spaces and Eulerian generating functions. Stud. Appl. Math. 49, 239–258MathSciNetMATHGoogle Scholar
  19. [19]
    Rota, G.-C. (1978): Hopf Algebra methods in combinatorics. In: Problèmes combinatoires et théorie des graphes. (Colloques Internationaux CNRS, vol. 260). Éditions du Centre National de la Recherche Scientifique, Paris, pp. 363–365Google Scholar
  20. [20]
    Rota, G.-C., Joni, S.A. (1979): Coalgebras and bialgebras in combinatorics. Stud. Appl. Math. 61, 93–139MathSciNetMATHGoogle Scholar
  21. [21]
    Rota, G.-C., Kung, J.P. (1984): The invariant theory of binary forms. Bull. Amer. Math. Soc. (N.S.) 10, 27–85MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    Rota, G.-C., Chen, W.Y.C. (1992): q-analogs of the inclusion-exclusion principle and permutations with restricted position. Discrete Math. 104, 7–22MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    Rota, G.-C. (1996): Report on the present state of combinatorics. Discrete Math. 153, 289–303MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Schur, I. (1981): Bermerkungen sur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen. J. Reine Angew. Math. 140, 1–28Google Scholar

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© Springer-Verlag Italia 2001

Authors and Affiliations

  • A. Lascoux

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