Catalan paths and Quasi-symmetric functions
refreshments at 3:45pm
We investigate the quotient ring $R$ of the ring of formal power
series $\Q[[x_1,x_2,\ldots]]$ over the closure of the ideal generated
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. [the equality is expected]
Speaker's Contact Info: bergeron(at-sign)mathstat.yorku.ca
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