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.