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  "body": "\n## Example\n\nLet $n \\ge 1$, and consider the ways to fully parenthesize a product $x_0 x_1 \\cdots x_n$ of $n + 1$ factors.\nThere are $C_n$ of them, where $C_n$ is the $n$-th [[Catalan number|0YWZJEABB2]].\nA rotation replaces a subexpression $(A B) C$ by $A (B C)$, where $A$, $B$, $C$ are subexpressions.\nThe Tamari order $T_n$ sets $p \\le q$ when $q$ can be reached from $p$ by a finite sequence of rotations.\n\nFor $n = 3$ there are $C_3 = 5$ bracketings of $abcd$.\nFrom the least element $((ab)c)d$ to the greatest element $a(b(cd))$ there are two routes:\n$$((ab)c)d \\to (ab)(cd) \\to a(b(cd)),$$\n$$((ab)c)d \\to (a(bc))d \\to a((bc)d) \\to a(b(cd)).$$\nThese are all the relations, so $T_3$ is the pentagon $N_5$, with a route of two steps beside a route of three steps.\n\n## Remarks\n\nEach $T_n$ is a [[lattice|WH4SFAACAS]].\nSince $T_3$ is the [[pentagon lattice|DHC8TZXGYS]], the Tamari lattices are not [[modular|QX7WZ92QAK]] in general.\nBracketings of $n + 1$ factors correspond to [[binary trees|0D7EZ10V2R]] with $n$ internal nodes, and a rotation is the tree rotation used in balanced search trees.\nThe Hasse diagram of $T_n$ is the graph of the associahedron.\n\n## Explanations\n\n### L1 human\n\nThere are several ways to put brackets into a long product, and moving brackets to the right is a single step.\nThe Tamari lattice records which bracketings can be reached from which by such steps.\nWith four factors there are five bracketings, arranged like a pentagon.\n",
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