Invitation is:

1. A document written or printed, or spoken words, /onveying the message by which one is invited.
2. The act of inviting; solicitation; the requesting of a person's company; as, an invitation to a party, to a dinner, or to visit a friend.
3. Allurement; enticement.
4. A request (spoken or written) to participate or be present or take part in something.


NOTE !!
- A written or spoken request asking someone to spend time with you socially or to come to a social event
The wedding invitations went out yesterday.
invitation to: Nobody in the office had received an invitation to the party.
- A request to someone to do something or go somewhere
invitation to do something: She has received an invitation to join a new environmental policy body.
decline/refuse an invitation: The senator declined our invitation to comment on the allegations.
standing/long-standing/open invitation (=with no fixed date): The former president has accepted a long-standing invitation to address the conference.
- A invitation or open invitation an action or event that makes something unpleasant more likely to happen
invitation to: He described the low wages paid to police officers as almost an invitation to corruption.
an (open) invitation to do something: The banks will take the minister’s comments as an open invitation to charge their customers higher rates.

Phrases

* At someone’s invitation/at the invitation of someone
if you do something at someone’s invitation, you do it when they ask or invite you
They came to Washington at the invitation of the president.

* By invitation (only)
available only to people who have been formally invited
a club with membership by invitation only

* Without invitation formal
without being asked or invited by anyone
When someone comes nearer without invitation, we feel threatened.