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Proceedings of the American Mathematical Society
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
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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.


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Additional Information

J.-C. Aval
Affiliation: Laboratoire A2X, Université Bordeaux 1, 351 cours de la Libération, 33405 Talence cedex, France
Email: aval@math.u-bordeaux.fr

N. Bergeron
Affiliation: Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3
Email: bergeron@mathstat.yorku.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06634-0
PII: S 0002-9939(02)06634-0
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