The enumeration of generalized Tamari intervals

• Published in: Transactions of the American Mathematical Society Vol. 369; no. 7; pp. 5219 - 5239
• Main Authors: PRÉVILLE-RATELLE, LOUIS-FRANÇOIS, VIENNOT, XAVIER
• Format: Journal Article
• Language: English
• Published: American Mathematical Society 01-07-2017
• Subjects: Combinatorics; Mathematics
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/7004

Let \overleftarrow {v} be the path obtained from v by reading the unit steps of v in reverse order, replacing the east steps by north steps and vice versa. We show that the poset Tam (v) is isomorphic to the dual of the poset Tam (\overleftarrow {v}). We do so by showing bijectively that the poset Tam (v) is isomorphic to the poset based on rotation of full binary trees with the fixed canopy v, from which the duality follows easily. This also shows that Tam (v) is a lattice for any path v. We also obtain as a corollary of this bijection that the usual Tamari lattice, based on Dyck paths of height n, can be partitioned into the (smaller) lattices Tam (v), where the v are all the paths on the square grid that consist of n-1 unit steps.