Maximum number of path for simple acyclic directed graph with start and end node

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jack1234
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Say given a simple acyclic directed graph with n nodes , which includes a starting node s0 and ending node e0 (i.e., a kripke structure without loop)
what is the maximum number of path from s0 to e0?
 

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The solution could be written in terms of powers of the (nilpotent) adjacency matrix.