This paper generalizes a construction of Geramita, Harima, and Shin
(Illinois J. Math. \textbf{45} (2001), 1--23). They give an
inductive description of a certain set of elements called $n$-type
vectors, and use these objects to prove various results about
Hilbert functions of sets of points. We extend their notation by
inductively describing the monomial ideals in $R$ and identifying
certain interesting subsets. We demonstrate that this new notation
is useful by using it to calculate multiplicity and the degree of
the Hilbert polynomial for quotients of Borel fixed ideals, and by
giving another proof of the result of Geramita, Harima, and Shin:
The set of $n$-type vectors is in bijective correspondence with all
Hilbert functions of finite length cyclic $R$-modules over the
polynomial ring $R=\poly{n}$, where $k$ is a field.