I just deleted CoverSutra which I purchased this morning off the App Store and it went from "Installed" to "Reinstall" or "Not Installed" (I can't really rememb

I just deleted CoverSutra which I purchased this morning off the App Store and it went from "Installed" to "Reinstall" or "Not Installed" (I can't really remember which) and I was able to reinstall it with one click and it came flying back into my dock. Pretty darn easy if you ask me. However, I'm not sure if the purchases are stored locally in the app itself/cloud? Say, If I just got a brand new machine, would my app list be stored in the cloud and ready for my viewing if I logged in with my account in the App Store? I'm not quite sure about that part though.