on double cosets with the trivial intersection property and kazhdan-lusztig cells in $s_n$

For a composition $lambda$ of $n$ our aim is to obtain reduced forms ‎for all the elements in the kazhdan-lusztig (right) cell containing ‎$w_{j(lambda)}$‎, ‎the longest element of the standard parabolic‎ ‎subgroup of $s_n$ corresponding to $lambda$‎. ‎we investigate how far this is possible to achieve by looking at‎ ‎elements of the form $w_{j(lambda)}d$‎, ‎where $d$ is a prefix of ‎an element of minimum length in a $(w_{j(lambda)},b)$ double coset‎ ‎with the trivial intersection property‎, ‎$b$ being a parabolic subgroup ‎of $s_n$ whose type is `dual' to that of $w_{j(lambda)}$‎.