Equivariant Hilbert Series of Monomial Orbits

The equivariant Hilbert series of an ideal generated by an orbit of a monomial under the action of the monoid $\mbox{Inc}(\mathbb{N})$ of strictly increasing functions is determined. This is used to find the dimension and degree of such an ideal. The result also suggests that the description of the denominator of an equivariant Hilbert series of an arbitrary $\mbox{Inc}(\mathbb{N})$-invariant ideal as given by Nagel and R\"omer is rather efficient.