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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

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