Weak order of permutations

In mathematics, the set of permutations on "n" items can be given the structure of a partial order, called the weak order of permutations. The weak order of permutations forms a lattice.

To define this order, consider the items being permuted to be the integers from 1 to "n", and let Inv("u") denote the set of inversions of a permutation "u" for the natural ordering on these items. That is, Inv("u") is the set of ordered pairs ("i", "j") such that 1 ≤ "i" < "j" ≤ "n" and "u"("i") > "u"("j"). Then, in the weak order, we define "u" ≤ "v" whenever Inv("u") ⊆ Inv("v").

The edges of the Hasse diagram of the weak order are given by permutations "u" and "v" such that "u < v" and such that "v" is obtained from "u" by interchanging two consecutive values of "u". These edges form a Cayley graph for the group of permutations that is isomorphic to the skeleton of a permutohedron.

The identity permutation is the minimum element of the weak order, and the permutation formed by reversing the identity is the maximum element.

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