: Mathématiques: Tamari lattice

mathematical analysis > combinatorics > algebraic combinatorics > Tamari lattice

algebra > combinatorics > algebraic combinatorics > Tamari lattice

algebra > abstract algebra > algebraic structure > lattice > Tamari lattice

Tamari lattice

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).
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Tamari_lattice)

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