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arXiv · math/0109147

Catalan paths, Quasi-symmetric functions and Super-Harmonic Spaces

Jean-Christophe Aval·Nantel Bergeron
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Abstract

We investigate the quotient ring $R$ of the ring of formal power series $\Q[[x_1,x_2,...]]$ over the closure of the ideal generated by non-constant quasi-\break 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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Jean-Christophe Aval, Nantel Bergeron. 2001-09-20. Catalan paths, Quasi-symmetric functions and Super-Harmonic Spaces. https://doi.org/10.1090/s0002-9939-02-06634-0

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