Pascal's Triangle and Hilbert Functions of Points
We define the Hilbert function of a finite set of points in projective
n-space. We then state the result of Geramita, Maroscia and Roberts
characterizes the Hilbert functions of points in terms of Macaulay's
O-sequences. For every possible Hilbert function, they construct a
points having that Hilbert function. Their constructions, called
k-configurations, have several properties of interest in themselves.
example, their Hilbert function, minimal free resolution and the degrees
of each point are easily determined. We generalize k-configurations
constructions which preserve these properties. Furthermore, since
O-sequences are defined only in terms of the binomial coefficients, we are
able to use a generalized Pascal's triangle to generalize Macaulay's
O-sequences. We use this generalization to characterize the Hilbert
functions of our genereralized k-configurations.