Billey, Jockusch & Stanley (1993) proved that in the finite symmetric group S n , a permutation is fully commutative if and only if it avoids the permutation pattern 321, that is, if and only if its one-line notation contains no three-term decreasing subsequence. In (Green 2002), this result was extended to affine permutations: an affine permutation u is fully commutative if and only if there do not exist integers i < j < k such that u ( i ) > u ( j ) > u ( k ) .