I think the best way to do this is to create a separate applications table. So as you've given the data, it would have two records, one for A and one for B.
So, with the addition of an Applications table and your existing table, which I'll call Objects, create a relationship from Applications to Objects (with a table occurrence called ObjectsParent) based on the ApplicationName as the match field. Create a self join relationship between Objects and itself with both Application and Program as the match fields. I'll call one of the "table occurrences" ObjectsParent and the other ObjectsChildren. Make sure that there's a primary key field in Objects that is set to auto-enter a serial number or some other method to ensure uniqueness. I'll call this ID.
So your relationship graph has three table occurrences:
Applications::Applicaiton = ObjectsParent::Application
ObjectsParent::Application = ObjectsChildren::Application, ObjectsParent::Program = ObjectsChildren::Program
Now create a calculation field in Objects, and calculating from the context of ObjectsParent, give it the following formula:
AppCount = Count( ObjectsChildren::ID )
Create a calculation field in Applications and calculating from the context of the table occurrence you used to relate it to ObjectsParent with the following formula:
AppCount = ObjectsParent::AppCount
The count field in Objects will have the same value for every object with the same application, so it doesn't matter which one you get this data from.
If you now view the data in Applications in list view, you can place the Applications::Application and Applications::AppCount fields on the layout and you should get what you've requested.