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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
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 Fall '08
 Propp
 Algebra, Combinatorics, Matrices, Counting

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