Catalan paths and quasi-symmetric functions
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- by J.-C. Aval and N. Bergeron PDF
- Proc. Amer. Math. Soc. 131 (2003), 1053-1062 Request permission
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
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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
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1053-1062
- MSC (2000): Primary 05E15, 05E10, 13P10, 13F30
- DOI: https://doi.org/10.1090/S0002-9939-02-06634-0
- MathSciNet review: 1948095