Below are Lynne M. Butler’s colorful slides on Enumeration and Partially Ordered Sets.
- Generating Functions and Inclusion-Exclusion
- Lattices and Möbius Functions
- Order Complexes and the Cohen-Macaulay Property
- Order Polynomials and Ehrhart Polynomials
The links to these slides on MSRI’s page are broken.
One of the slides from Lattices and Möbius Functions is shown below.
Haverford physician Joel Lowenthal asked about the topological fact that the largest number of pieces into which you can slice a pie using n straight cuts equals 1 + ( 1 + 2 + … + n ) = 1 + n + n(n-1)/2 . Zaslavsky’s result is a generalization of this fact.
To explain Zaslavsky’s result, I’ll describe how to draw the lattice at the lower right from the cuts pictured at the lower left. The 4 cuts, H1, H2, H3, and H4, correspond to the 4 red dots in the lattice labelled by the green number -1. The 4 red dots just above these correspond to the 4 points of intersection of the cuts. The rightmost of these red dots is connected by 3 red line segments to the red dots corresponding to H2, H3, and H4 because these cuts intersect at the same point. Notice that the green number 2 on this rightmost red dot is 1 fewer than the number 3 of these line segments. Zaslavsky’s result is that the total number of pieces into which the plane is cut equals the sum of the absolute values of the green numbers, and that the number of bounded pieces equals the absolute value of the sum of the green numbers.