Title:
Catalan paths and quasi-symmetric functions
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-symmetric functions. We show that a Hilbert (topological) linear 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] |