How to compute number of possible paths in directed Traveling Salesman problem? If we have directed modification of TSP, so that some routes are possible in one direction, given directed graph, could you compute number of possible tours?

What if we have supercomputer, does it make dTSP more feasible?

edit:The number of different Hamiltonian paths in undirected graph is (n−1)!/2 in fully directed version it is (n−1)!, exact number could be computed from adjacency matrix

yes but if i delete city or two from connection like this example will not be fully and this is my case:

1 to 2,3

2 to 3

3 to 1,2,4,5

4 to 2,5,1

5 to 1

so i convert dTSP to simple graph my problem is to know how many paths can i create by my edited dTSP graph so
if i have

1 to 2,3

2 to 1,3

3 to 1,2

i will have number 3!= 6 paths
but if i edit it to be like that 

1 to 2,3

2 to 3

3 to 1

i will have 3 paths only

i found that if one city is delete from connections can cut paths maybe by double as i try and error with small numbers of cities so i want to know if i possibly can calculate number of paths that created by this given connections
The way you have written your question makes it very difficult to understand, but let me try to answer one of your points as simply as I can.

Using a "super computer" (however you might define that) will not help us to make significant progress on the travelling salesman problem. The reason is that the time taken to solve a travelling salesman problem increases exponentially with the number of cities.

So suppose we can solve problems with up to 50 cities in a reasonable time on an "ordinary" computer (I have made this number up as an illustration). But let's also suppose that the running time of our most efficient algorithm is $O(2^n)$ - in other words, every time we add a city the time needed to find a solution doubles.

If we had a super computer that was a thousand times faster than an ordinary computer then it could solve problems with $60$ cities in the same time as an ordinary computer took for $50$ cities. This is because adding $10$ cities increases the problem difficulty by a factor of $2^{10} \approx 1000$.

So the super computer gives us a small improvement. But to solve  problems with $100$ cities in a reasonable time we would need a computer that was a million million ($10^{12}$) times faster again.