  We prove a formula connecting the number of unipotent conjugacy classes in a
maximal parabolic subgroup of a finite general linear group with the numbers of
unipotent conjugacy classes in various parabolic subgroups in smaller
dimensions. We generalise this formula and deduce a number of corollaries; in
particular, we express the number of conjugacy classes of unitriangular
matrices over a finite field in terms of the numbers of unipotent conjugacy
classes in maximal parabolic subgroups over the same field. We show how the
numbers of unipotent conjugacy classes in parabolic subgroups of small
dimensions may be calculated.
