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- Tamari_lattice abstract "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.The Tamari lattice can also be described in several other equivalent ways:It is the poset of sequences of n integers a1, ..., an, ordered coordinatewise, such that i ≤ ai ≤ n and if i ≤ j ≤ ai then aj ≤ ai (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.".
- Tamari_lattice thumbnail Tamari_lattice.svg?width=300.
- Tamari_lattice wikiPageExternalLink fasc4a.ps.gz.
- Tamari_lattice wikiPageID "20353278".
- Tamari_lattice wikiPageRevisionID "583010221".
- Tamari_lattice authorlink "Dov Tamari".
- Tamari_lattice first "Dov".
- Tamari_lattice hasPhotoCollection Tamari_lattice.
- Tamari_lattice last "Tamari".
- Tamari_lattice year "1962".
- Tamari_lattice subject Category:Lattice_theory.
- Tamari_lattice comment "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)).".
- Tamari_lattice label "Tamari lattice".
- Tamari_lattice label "Treillis de Tamari".
- Tamari_lattice sameAs Treillis_de_Tamari.
- Tamari_lattice sameAs m.04_0n1v.
- Tamari_lattice sameAs Q7680890.
- Tamari_lattice sameAs Q7680890.
- Tamari_lattice wasDerivedFrom Tamari_lattice?oldid=583010221.
- Tamari_lattice depiction Tamari_lattice.svg.
- Tamari_lattice isPrimaryTopicOf Tamari_lattice.