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The Tamari lattice Tn was originally defined on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Although in several related lattices, the number of maximal chains is known, quoting Knuth, "The enumeration of such paths in Tamari lattices remains mysterious." The lengths of maximal chains vary over a great range. In this paper, we focus on the chains with maximum length in these lattices. We establish a bijection between the maximumdoi:10.1090/s0002-9939-2014-12069-7 fatcat:gqqqloslzndutnv3fulpmhx2qq