On double cosets with the trivial intersection property and Kazhdan-Lusztig cells in S n

For a composition λ of n our aim is to obtain reduced forms for all the elements in the Kazhdan-Lusztig (right) cell containing w J(λ), the longest element of the standard parabolic subgroup of S n corresponding to λ. We investigate how far this is possible to achieve by looking at elements of the form w J(λ) d, where d is a prefix of an element of minimum length in a (W J(λ), B) double coset with the trivial intersection property, B being a parabolic subgroup of S n whose type is `dual'' to that of W J(λ).