Let $n$ be a positive integer.
Consider an ordered set $S_n = [1,2,3,...,n]$ where the $j$ th element from the left equals $j$.
Now consider a function defined on $S_n$ as a permutation of that set.
$$ f(S_n) = P(S_n) $$
Now iterating the function $f$ means iteratively rearranging the set like the permution $f$ does.
In notation
$$f^m(S_n) = f( f^{m-1}(S_n)) = P ( f^{m-1}(S_n)) = P^m(S_n)$$
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Let $f$ be a "good" function IFF it does not map any $a$ in $S_n$ to it same position.
Let $G(n,b) $ denote the cardinality of functions $f$ for $S_n$ such that $f,f^2,...f^b$ are all good functions.
What is known about $G(n,b)$ ?
Can it be given in closed form ?
Does it have an integral representation ?
I can not help thinking about binomial and euler numbers but im confused. Maybe this is wrong, maybe not.
This seems to me like a very simple question, so maybe the answer is very well known and the result is named ?
I think a good way to look at this is through the cycle representation of the permutation. What we want is the number of permutations such that each cycle has size at least $b+1$.
One possible way to count them is to run through all number partitions of $n$ into parts of size at least $b+1$ (so running through all possible cycle types). For each cycle type we can find the number of permutations with this simple formula.
Iterating through all possible number partitions can be done rather easily if we order them increasingly and go through them lexicographically.