Tamari lattice

Example

Let $n \ge 1$, and consider the ways to fully parenthesize a product $x_0 x_1 \cdots x_n$ of $n + 1$ factors. There are $C_n$ of them, where $C_n$ is the $n$-th Catalan number. A rotation replaces a subexpression $(A B) C$ by $A (B C)$, where $A$, $B$, $C$ are subexpressions. The Tamari order $T_n$ sets $p \le q$ when $q$ can be reached from $p$ by a finite sequence of rotations.

For $n = 3$ there are $C_3 = 5$ bracketings of $abcd$. From the least element $((ab)c)d$ to the greatest element $a(b(cd))$ there are two routes: $$((ab)c)d \to (ab)(cd) \to a(b(cd)),$$ $$((ab)c)d \to (a(bc))d \to a((bc)d) \to a(b(cd)).$$ These are all the relations, so $T_3$ is the pentagon $N_5$, with a route of two steps beside a route of three steps.