Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Catalan paths and quasi-symmetric functions

Authors: J.-C. Aval and N. Bergeron
Journal: Proc. Amer. Math. Soc. 131 (2003), 1053-1062
MSC (2000): Primary 05E15, 05E10, 13P10, 13F30
Published electronically: July 26, 2002
MathSciNet review: 1948095
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the quotient ring $R$ of the ring of formal power series $\mathbb{Q} [[x_1,x_2,\ldots]]$ over the closure of the ideal generated by non-constant quasi-symmetric functions. We show that a Hilbert basis of the quotient is naturally indexed by Catalan paths (infinite Dyck paths). We also give a filtration of ideals related to Catalan paths from $(0,0)$ and above the line $y=x-k$. We investigate as well the quotient ring $R_n$ of polynomial ring in $n$ variables over the ideal generated by non-constant quasi-symmetric polynomials. We show that the dimension of $R_n$ is bounded above by the $n$th Catalan number.

References [Enhancements On Off] (What's this?)

  • 1. E. ARTIN, Galois Theory, Notre Dame Mathematical Lecture 2 (1944), Notre Dame, IN. MR 5:225c; reprint MR 98k:12001, MR 42:234
  • 2. J.-C. AVAL, F. BERGERON AND N. BERGERON, Ideal of quasi-symmetric functions and super-covariant polynomials for $s_n$, to appear (Math. co 10202071).
  • 3. J.-C. AVAL, F. BERGERON, N. BERGERON AND A. GARSIA, Ideals of Quasi-symmetric polynomials and related varieties, in preparation.
  • 4. F. BERGERON, N. BERGERON, A. GARSIA, M. HAIMAN AND G. TESLER, Lattice Diagram Polynomials and Extended Pieri Rules, Adv. Math. 142 (1999), 244-334. MR 2000h:05231
  • 5. F. BERGERON, A. GARSIA AND G. TESLER, Multiple Left Regular Representations Generated by Alternants, J. of Comb. Th., Series A 91, 1-2 (2000), 49-83. MR 2002c:05158
  • 6. N. BERGERON, S. MYKYTIUK, F. SOTTILE, AND S. VAN WILLIGENBURG, Pieri Operations on Posets,
    J. of Comb. Theory, Series A 91 (2000), 84-110.
  • 7. C. DE CONCINI AND C. PROCESI, Symmetric functions, conjugacy classes and the flag variety, Invent. Math. 64 (1981), 203-230. MR 82m:14030
  • 8. D. COX, J. LITTLE AND D. O'SHEA, Ideals, Varieties, and Algorithms, Springer-Verlag, New-York, 1992. MR 93j:13031; second edition MR 97h:13024
  • 9. A. M. GARSIA AND M. HAIMAN, A graded representation model for Macdonald's polynomials, Proc. Nat. Acad. Sci. U.S.A. 90 (1993), no. 8, 3607-3610. MR 94b:05206
  • 10. I. GELFAND, D. KROB, A. LASCOUX, B. LECLERC, V. RETAKH, AND J.-Y. THIBON, Noncommutative symmetric functions, Adv. in Math. 112 (1995), 218-348. MR 96e:05175
  • 11. I. GESSEL, Multipartite ${P}$-partitions and products of skew Schur functions, in Combinatorics and Algebra (Boulder, Colo., 1983), C. Greene, ed., vol. 34 of Contemp. Math., AMS, 1984, pp. 289-317. MR 86k:05007
  • 12. M. HAIMAN, Hilbert schemes, polygraphs, and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), 941-1006. MR 2002c:14008
  • 13. F. HIVERT, Hecke algebras, difference operators, and quasi-symmetric functions, Adv. Math. 155 (2000), 181-238. CMP 2001:04
  • 14. I. MACDONALD, Symmetric Functions and Hall Polynomials, Oxford Univ. Press, 1995,
    second edition. MR 96h:05207
  • 15. C. MALVENUTO AND C. REUTENAUER, Duality between quasi-symmetric functions and the Solomon descent algebra, Journal of Algebra 177 (1995), 967-982. MR 97d:05277
  • 16. R. STANLEY, Enumerative Combinatorics, Vol. 1, Wadsworth and Brooks/Cole, 1986. MR 87j:05003; corr. reprint MR 98a:05001
  • 17. R. STANLEY, Enumerative Combinatorics Vol. 2, no. 62 in Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1999.
    Appendix 1 by Sergey Fomin. MR 2000k:05026
  • 18. R. STEINBERG, Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc. 112 (1964), 392-400. MR 29:4807

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 05E15, 05E10, 13P10, 13F30

Retrieve articles in all journals with MSC (2000): 05E15, 05E10, 13P10, 13F30

Additional Information

J.-C. Aval
Affiliation: Laboratoire A2X, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex, France

N. Bergeron
Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3

Received by editor(s): October 16, 2001
Received by editor(s) in revised form: November 8, 2001
Published electronically: July 26, 2002
Additional Notes: The second author was supported in part by NSERC, PREA and CRC
Communicated by: John R. Stembridge
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society