## Pascal's Triangle and Hilbert Functions of Points

**Sindi Sabourin**

(Queens University)

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
which

characterizes the Hilbert functions of points in terms of Macaulay's

O-sequences. For every possible Hilbert function, they construct a
set of

points having that Hilbert function. Their constructions, called

k-configurations, have several properties of interest in themselves.
For

example, their Hilbert function, minimal free resolution and the degrees

of each point are easily determined. We generalize k-configurations
to

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.