No email notifications when Mail.app isn't running? On Mountain Lion, I don't get email notifications in Notification Center when Mail.app isn't running.

This is not the expected behaviour (it is inconsistent with the Mail.app behaviour on iOS) and makes the notification feature redundant (Mail.app shows a new email badge and also has an email-snippet view).

Is there a setting or third-party solution to fix this?

UPDATE:
When I'm interacting with email I want to be able to command+tab.
This is very handy when composing a message that consists of copy-pasted content from other applications.
I also use the "drag a file to the dock icon" functionality to create a e-mail with an attachment.

But when I'm not interacting with email (98% of the time) I don't wan't Mail.app cluttering the application-switcher but do want to be notified of new email (via either a notification or a badge).
I can only say that this is true. Mail.app has to be open to get notifications.


  A workaround: Open Mail, and close it with the red dot (or command+w). So the blue
  point under the icon in the Dock is still there. So you have no window
  open but mail is still getting emails.


There is no app or anything there yet to fix this "problem".