Weak order of permutations

Example

Let $n \ge 1$ and let $S_n$ be the symmetric group of permutations of $\set{1, \ldots, n}$, written in one-line notation $w = w(1) w(2) \cdots w(n)$. The inversion set $\mathrm{Inv}(w)$ is the set of pairs of values $(a, b)$ with $a < b$ such that $b$ stands left of $a$ in $w$. The weak order on $S_n$ is defined by $u \le w$ when $\mathrm{Inv}(u) \subseteq \mathrm{Inv}(w)$. It is a partial order, and it is a lattice. The covering relations are the moves that swap two adjacent entries $a < b$ standing as $\ldots a b \ldots$. Such a move adds the single pair $(a, b)$ to the inversion set. The least element is the identity $12 \cdots n$, and the greatest element is the reversal $n \cdots 21$.

Check for $n = 3$. The inversion sets are $\mathrm{Inv}(123) = \emptyset$, $\mathrm{Inv}(213) = \set{(1,2)}$, $\mathrm{Inv}(132) = \set{(2,3)}$, $\mathrm{Inv}(231) = \set{(1,2), (1,3)}$, $\mathrm{Inv}(312) = \set{(1,3), (2,3)}$, and $\mathrm{Inv}(321)$ has all three pairs. So the weak order is the two chains $123 < 213 < 231 < 321$ and $123 < 132 < 312 < 321$. The elements $213$ and $312$ are incomparable, since $(1,2) \notin \mathrm{Inv}(312)$.