The Bruhat order on S ~ ~ n has the following combinatorial realization. If u is an affine permutation and i and j are integers, define u [ i , j ] to be the number of integers a such that a ≤ ≤ i and u ( a ) ≥ ≥ j . (For example, with u = [ 2 , 0 , 4 ] ∈ ∈ S ~ ~ 3 , one has u [ 3 , 1 ] = 3 : the three relevant values are a = 0 , 1 , 3 , which are respectively mapped by u to 1, 2, and 4.) Then for two affine permutations u, v, one has that u ≤ ≤ v in Bruhat order if and only if u [ i , j ] ≤ ≤ v [ i , j ] for all integers i, j.