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

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In a simple acyclic directed graph with n nodes, the maximum number of paths from a starting node s0 to an ending node e0 can be determined using the powers of the nilpotent adjacency matrix. The adjacency matrix captures the connections between nodes, and its powers can reveal the number of distinct paths of varying lengths. Specifically, the entries in the matrix indicate the number of paths of a certain length between nodes. The maximum number of paths is influenced by the structure of the graph and the arrangement of nodes. Understanding this relationship is crucial for analyzing path counts in directed acyclic graphs.
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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.
 

Thread 'Hypothesis testing: Defining H0, HA hypotheses so that ( H_A)_A' makes sense'

The standard _A " operator" maps a Null Hypothesis Ho into a decision set { Do not reject:=1 and reject :=0}. In this sense ( HA)_A , makes no sense. Since H0, HA aren't exhaustive, can we find an alternative operator, _A' , so that ( H_A)_A' makes sense? Isn't Pearson Neyman related to this? Hope I'm making sense. Edit: I was motivated by a superficial similarity of the idea with double transposition of matrices M, with ## (M^{T})^{T}=M##, and just wanted to see if it made sense to talk...
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