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

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

2001-09-20

arxiv.org/abs/math/0109147v1

Proc. Amer. Math. Soc. 131 (2003), no. 4, 1053-1062

Mathematics

Combinatorics

14 pages

Scientific paper

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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