Imagine a number $n$ of vertices. How many different graphs can be constructed given the following rules:
- No segment of the graph can be disconnected, this means there must be a path between any pair of vertices.
- All connections must be directed with an arrow.
- A vertex with no inputs has a value of 1.
- If two vertex are connected, the consequent one must have a bigger value than the first one. (This also means there cannot be any cycles)
- If there is a path moving only forward between two vertices, there cannot be a direct shortcut connection between them.
Is there an algorithm I can use to make sure I have count them all?