She attended the fund-raiser 'at the invitation of' the committee chairperson. [=she attended because she was invited by the committee chairperson]
You have a 'standing/open invitation' [=you are always welcome; you are always invited] to visit us any time.
The event is 'by invitation only'. [=only people who have been invited to the event can attend it]
(..) and I don't have to tell you, that,you know,a clean colon is just one less thing to worry about. Leonard,I'm no expert here, but I believe in the context of a luncheon invitation, you might want to skip the reference to bowel movements. Oh,you're inviting me over to eat? Uh... yes. (..)2007 TvShow: The Big Bang Theory Title: Pilot Season: 1 Episode 1