  We introduce the active partition of the ground set of an oriented matroid
perspective (or quotient, or strong map) on a linearly ordered ground set. The
reorientations obtained by arbitrarily reorienting parts of the active
partition share the same active partition. This yields an equivalence relation
for the set of reorientations of an oriented matroid perspective, whose classes
are enumerated by coefficients of the Tutte polynomial, and a remarkable
partition of the set of reorientations into boolean lattices, from which we get
a short direct proof of a 4-variable expansion formula for the Tutte polynomial
in terms of orientation activities. This formula was given in the last
unpublished preprint by Michel Las Vergnas; the above equivalence relation and
notion of active partition generalize a former construction in oriented
matroids by Michel Las Vergnas and the author; and the possibility of such a
proof technique in perspectives was announced in the aforementioned preprint.
We also briefly highlight how the 5-variable expansion of the Tutte polynomial
in terms of subset activities in matroid perspectives comes in a similar way
from the known partition of the power set of the ground set into boolean
lattices related to subset activities (and we complete the proof with a
property which was missing in the literature). In particular, the paper applies
to matroids and oriented matroids on a linearly ordered ground set.
