  The theory of q-analogs develops many combinatorial formulas for finite
vector spaces over a finite field with q elements--all in analogy with formulas
for finite sets (which are the special case of q=1). A direct-sum decomposition
of a finite vector space is the vector space analogue of a set partition. This
paper develops the formulas for the number of direct-sum decompositions that
are the q-analogs of the formulas for: (1) the number of set partitions with a
given number partition signature; (2) the number of set partitions of an
n-element set with m blocks (the Stirling numbers of the second kind); and (3)
for the total number of set partitions of an n-element set (the Bell numbers).
