Concrete mathematics : a foundation for computer science

36 SUMS
instead of summing over all integers j and k. These variations come in two
flavors, vanilla and rocky road. First, the vanilla version:
(2.29)
This is just another way to write (2.27), since the Iversonian [j E J, kE K]
factors into [j E J] [k E K]. The vanilla-flavored law applies whenever the ranges
of j and k are independent of each other.
The rocky-road formula for interchange is a little trickier. It applies when
the range of an inner sum depends on the index variable of the outer sum:
x t ai,k = x t ai,k.
jEJ kEK(j)
M’K’ iEJ’(k)
Here the sets J, K(j), K’, and J’(k) must be related in such a way that
[jEJl[kEK(j)] = [kEK’l[jEJ’(k)].
(2.30)
A factorization like this is always possible in principle, because we can let
J = K’ be the set of all integers and K(j) = J’(k) be the basic property P(j, k)
that governs a double sum. But there are important special cases where the
sets J, K(j), K’, and J’(k) have a simple form. These arise frequently in
applications. For example, here’s a particularly useful factorization:
[16j