Tamari lattice

Tamari lattice of order 4

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).
• 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.

Notation

The Tamari lattice of the C_n groupings of n+1 objects is called T_n, but the corresponding associahedron is called K_n+1.

In The Art of Computer Programming T₄ is called the Tamari lattice of order 4 and its Hasse diagram K₅ the associahedron of order 4.

References

• Chapoton, F. (2005), "Sur le nombre d'intervalles dans les treillis de Tamari" (in French), Séminaire Lotharingien de Combinatoire 55 (55): 2368, Bibcode: 2006math......2368C .
• Csar, Sebastian A.; Sengupta, Rik; Suksompong, Warut (2014), "On a Subposet of the Tamari Lattice", Order 31 (3): 337–363, doi:10.1007/s11083-013-9305-5 .
• Early, Edward (2004), "Chain lengths in the Tamari lattice", Annals of Combinatorics 8 (1): 37–43, doi:10.1007/s00026-004-0203-9 .
• "Problèmes d'associativité: Une structure de treillis finis induite par une loi demi-associative" (in French), Journal of Combinatorial Theory 2 (3): 215–242, 1967, doi:10.1016/S0021-9800(67)80024-3 .
• Geyer, Winfried (1994), "On Tamari lattices", Discrete Mathematics 133 (1–3): 99–122, doi:10.1016/0012-365X(94)90019-1 .
• Huang, Samuel (1972), "Problems of associativity: A simple proof for the lattice property of systems ordered by a semi-associative law", Journal of Combinatorial Theory, Series A 13: 7–13, doi:10.1016/0097-3165(72)90003-9 .
• "Draft of Section 7.2.1.6: Generating All Trees", The Art of Computer Programming, IV, 2005, p. 34, http://www-cs-faculty.stanford.edu/~knuth/fasc4a.ps.gz .
• "The algebra of bracketings and their enumeration", Nieuw Archief voor Wiskunde, Series 3 10: 131–146, 1962 .