# Weak order of permutations

In mathematics, the symmetric group,*S*, has a poset structure given by the

_{n}**weak order of permutations**, given by

*u≤v*if Inv(

*u*) is a subset of Inv(

*v*). Here Inv(

*u*) is the set of inversions of

*u*, defined as the set of ordered pairs (

*i*,

*j*) with

- 1 ≤
*i*<*j*≤*n*

*u*(*i*) >*u*(*j*).

*u*and

*v*such that

*u < v*

*v*is obtained from

*u*by interchanging two consecutive values of

*u*.

The identity permutation is the minimum element of *S _{n}* and the permutation

*(n n-1 ... 1)*is the maximum element.

Moreover, *S _{n}* is a lattice with this order.

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