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Your poset is either the same or closely related to the one in http://www.fh-bingen.de/fileadmin/user_upload/Lehrende/Winkel_Rudolf/06.pdf, which is a lattice.
(Figure added by J.O'Rourke.)
Your poset is either the same or closely related to the one in http://www.fh-bingen.de/fileadmin/user_upload/Lehrende/Winkel_Rudolf/06.pdf which is a lattice.