This preview shows pages 1–2. Sign up to view the full content.
homework
Any questions about the homework or the material?
TODAY:
Paths in DAGs
Counting paths by multiplying matrices
Domino tilings again
Weighted enumeration
A finite DAG (directed acyclic graph) is a finite set of vertices
with arcs connecting one vertex to another, such that
two vertices are linked by only finitely many arcs, and
there are no infinite cycles.
In any finite DAG, there are
only finitely many paths connecting any two vertices.
Write N(x,y) as the number of paths from x to y.
Note N(x,x) = 1 for all x.
Theorem: Suppose G is a finite DAG,
x,y are vertices of G,
and B is a set of vertices with the property that
every path from x to y passes through
exactly one vertex in B.
Then N(x,y) = sum_{b in B} N(x,b) N(b,y).
More generally,
suppose A is the set of sources
(define)
in G
and C is the set of sinks
(define)
in G,
and that B has the property that
every path from A to C passes through
exactly one vertex in B.
Define the transfer matrix N(A,B) as the matrix
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 Propp
 Algebra, Combinatorics, Matrices, Counting

Click to edit the document details