Let $n\geq4$ how many permutations $\pi$ of $S_n$ has the property that $1,2,3$ appear in the same cycle of $\pi$.... Let $n\geq4$ how many permutations $\pi$ of $S_n$ has the property that $1,2,3$ appear in the same cycle of $\pi$,while $4$ appears in a different cycle of $\pi$ from $1,2,3$?
My attempt:For $n=4$ I have found two such permutations $(123)(4)$ and $(132)(4)$
For $n=5$ I have found four such permutations $(123)(45),(132)(45),(123)(4)(5),(132)(4)(5)$
For $n=6$ I have found $12$ such permutations.
$(123)(456),(132)(456),(123)(465),(132)(465),(123)(45)(6),(132)(45)(6),(123)(46)(5),(132)(46)(5),(123)(56)(4),(132)(56)(4),(123)(4)(5)(6),(132)(4)(5)(6)$
My problem:How we can generally solve this?What will be the general formula for number of such permutations? 
Edit:
For $n=5$ we also have the permutations $(1235)(4),(1325)(4),(1532)(4)$ and similarly these are 6.So in $S_5$ we have $4+6=10$ such permutations.

For $n\ge 4$, let $f(n)$ be the number of permutations in $S_n$ in which $1,2,3$ are all in the same cycle and $4$ is in a different cycle.
To go from $f(n)$ to $f(n+1)$ observe that for each desirable permutation in $S_n$ we can build $n+1$ desirable permutations in $S_{n+1}$ by inserting $n+1$ somewhere in each existing cycle ($n$ ways) or by mapping $n+1$ to itself.
For example, from the desirable permutation $(1532)(46)(7)$ in $S_7$, we have correspondingly $8$ desirable permutations ins $S_8$:
$(18532)(46)(7)$
$(15832)(46)(7)$
$(15382)(46)(7)$
$(15328)(46)(7)$
$(1532)(486)(7)$
$(1532)(468)(7)$
$(1532)(46)(78)$
$(1532)(46)(7)(8)$
So in general $f(n+1)=(n+1)f(n)$
This also implies that the ratio of desirable permutations to total permutations remains constant.
You already noted that $f(4)=2$.
