The Smurfs

How many ways of smurfing ?

A smurf A could smurf a smurf B without B smurfes A
A smurf could sometimes smurf himself
A smurf could smurf several smurfs, or be smurfed by several smurfs.

With only one smurf, there are two ways of smurfing :
- the smurf doesn't smurf
- the smurf smurfes himself

With two smurfs, there is even 16 ways of smurfing, for instance :
- no smurf smurf
- A smurfes himself
- A smurfes himself and B smurfes himself
- A surfes himself and A smurfes B
- A smurfes B and B smurfes A
- A smurfes himself, B smurfes himself, A smurfes B and B smurfes A
- ...

With 3 smurfs, how many ways of smurfing ?   Solution
With 4 smurfs ?   Solution   With n smurfs ?   Solution   Details

Restrictions

We know that a smurf can't smurf more than p smurfs, and a smurf can't be smurfed by more than q smurfs.
How many ways of smurfing remains ?
With two smurfs, p=1, q=1 ?  p=1, q=2 ? Solution
With 3 smurfs, p,q=1,2,3 ? Solution
With n smurfs ?   Solution   Details

 

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