<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume11 Issue1" %>
Parking Functions and Descent Algebras
Jean-Christophe Novelli, Jean-Yves Thibon
Institut Gaspard Monge, Université de Marne-la-Vallée, 5 Boulevard Descartes, Champs-sur- Marne, 77454 Marne-la-Vallée cedex 2, France
{novelli, jyt}@univ-mlv.fr
Annals of Combinatorics 11 (1) p.59-68 March, 2007
AMS Subject Classification: 05E05, 16W30
Abstract:
We show that the notion of parkization of a word, a variant of the classical standardization, allows us to introduce an internal product on the Hopf algebra of parking functions. Its Catalan subalgebra is stable under this operation and contains the descent algebra as a left ideal.
Keywords: parking functions, descent algebras, quasi-symmetric functions

References:

1. G. Duchamp, F. Hivert, and J.-Y. Thibon, Noncommutative symmetric functions VI: free quasi-symmetric functions and related algebras, Internat. J. Algebra Comput. 12 (2002) 671- 717.

2. I.M. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V.S. Retakh, and J.-Y. Thibon, Noncommutative symmetric functions, Adv. Math. 112 (1995) 218-348.

. I. Gessel, Multipartite P-partitions and inner product of skew Schur functions, Contemp. Math. 34 (1984) 289-301.

4. F. Hivert, J.-C. Novelli, and J.-Y. Thibon, Commutative Hopf algebras of permutations and trees, preprint, arXiv:math.CO/0502456.

5. D. Krob, B. Leclerc, and J.-Y. Thibon, Noncommutative symmetric functions II: transformations of alphabets, Internat. J. Algebra Comput. 7 (1997) 181-264.

6. J.-L. Loday and M. Ronco, Hopf algebra of the planar binary trees, Adv. Math. 139 (1998) 293-309.

7. I.G. Macdonald, Symmetric Functions and Hall Polynomials, Second Edition, Oxford University Press, 1995.

8. C. Malvenuto and C. Reutenauer, Duality between quasi-symmetric functions and the Solomon descent algebra, J. Algebra 177 (3) (1995) 967-982.

9. J.-C Novelli and J.-Y. Thibon, A Hopf algebra of parking functions, Proc. FPSAC/SFCA 2004, Vancouver.

10. F. Patras, L'algèbre des descentes d'une bigèbre graduée, J. Algebra 170 (2) (1994)547-566.

11. F. Patras and C. Reutenauer, Lie representations and an algebra containing Solomon's, J. Algebraic Combin. 16 (2002) 301-314.

12. C. Reutenauer, Free Lie Algebras, London Math. Soc. Monogr. (N.S.) 7, Oxford University Press, New York, 1993.

13. M. Schocker, Lie idempotent algebras, Adv. Math. 175 (2) (2003) 243-270.

14. L. Solomon, A Mackey formula in the group ring of a Coxeter group, J. Algebra 41 (2) (1976) 255-264.

15. J.-Y. Thibon and B.C.V. Ung, Quantum quasi-symmetric functions and Hecke algebras, J. Phys. A 29 (22) (1996) 7337-7348.