Ideals and quotients of B-quasisymmetric functions

Abstract

The space $QSy{m}_n(B)$ of $B$-quasisymmetric polynomials in 2 sets of $n$ variables was recently studied by Baumann and Hohlweg. The aim of this work is a study of the ideal $<QSy{m}_n(B{)}^{+}>$ generated by $B$-quasisymmetric polynomials without constant term. In the case of the space $QSy{m}_n$ of quasisymmetric polynomials in 1 set of $n$ variables, Aval, Bergeron and Bergeron proved that the dimension of the quotient of the space of polynomials by the ideal $<QSy{m}_n^{+}>$ is given by Catalan numbers $C_n=\frac{1}{n+1}\left(\genfrac{}{}{0ex}{}{2n}{n}\right)$. In the case of $B$-quasisymmetric polynomials, our main result is that the dimension of the analogous quotient is equal to $\frac{1}{2n+1}\left(\genfrac{}{}{0ex}{}{3n}{n}\right)$, the numbers of ternary trees with $n$ nodes. The construction of a Gröbner basis for the ideal, as well as of a linear basis for the quotient are interpreted by a bijection with lattice paths. These results are finally extended to $p$ sets of variables, and the dimension is in this case $\frac{1}{pn+1}\left(\genfrac{}{}{0ex}{}{(p+1)n}{n}\right)$, the numbers of $p$-ary trees with $n$ nodes.

Cited by in corpus (1)

• Multivariate Fuss-Catalan numbers