In mathematics, a Tamari lattice, introduced by Dov Tamari (1962), is a partially ordered set in which the elements consist of different ways of grouping a sequence of objects into pairs using parentheses; for instance, for a sequence of four objects abcd, the five possible groupings are ((ab)c)d, (ab)(cd), (a(bc))d, a((bc)d), and a(b(cd)). Each grouping describes a different order in which the objects may be combined by a binary operation; in the Tamari lattice, one grouping is ordered before another if the second grouping may be obtained from the first by only rightward applications of the associative law (xy)z = x(yz). For instance, applying this law with x = a, y = bc, and z = d gives the expansion (a(bc))d = a((bc)d), so in the ordering of the Tamari lattice (a(bc))d ≤ a((bc)d).

In this partial order, any two groupings g₁ and g₂ have a greatest common predecessor, the meet g₁ ∧ g₂, and a least common successor, the join g₁ ∨ g₂. Thus, the Tamari lattice has the structure of a lattice. The Hasse diagram of this lattice is isomorphic to the graph of vertices and edges of an associahedron. The number of elements in a Tamari lattice for a sequence of n + 1 objects is the nth Catalan number C_n.

The Tamari lattice can also be described in several other equivalent ways:

• It is the poset of sequences of n integers a₁, ..., a_n, ordered coordinatewise, such that i ≤ a_i ≤ n and if i ≤ j ≤ a_i then a_j ≤ a_i (Huang & Tamari 1972).
• It is the poset of binary trees with n leaves, ordered by tree rotation operations.
• It is the poset of ordered forests, in which one forest is earlier than another in the partial order if, for every j, the jth node in a preorder traversal of the first forest has at least as many descendants as the jth node in a preorder traversal of the second forest (Knuth 2005).
• It is the poset of triangulations of a convex n-gon, ordered by flip operations that substitute one diagonal of the polygon for another.