For a composition lambda of n we consider the Kazhdan-Lusztig cell in the symmetric group Sn containing the longest element of the standard parabolic subgroup of Sn associated to lambda. In this paper, we extend some of the ideas and results in [Beitrage zur Algebra und Geometrie, 59(3) (2018) 523-547]. In particular, by introducing the notion of an ordered k-path, we are able to obtain alternative explicit descriptions for some additional families of cells associated to compositions. This is achieved by first determining the rim of the cell, from which reduced forms for all the elements of the cell are easily obtained.
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