Should I say "Allahu Akbar" for the Ihram and then pat the latecomer on the shoulder, or should I pat him and then say "Allahu Akbar" for the Ihram?

The preponderant view is that it is valid for a latecomer (masbuq) to lead others in prayer after his Imam has given the salutation. It is not a condition for the validity of the congregational prayer that the follower (mu'tam) be aware of the Imam's intention to lead, because the intention of leading is recommended, not obligatory. There is no specific prescribed method for informing him, and it is permissible to inform him by any means. If the person joining the prayer wishes to alert the latecomer, this should be done before the opening takbir (takbirat al-ihram (definition)).