Concrete mathematics : a foundation for computer science

2.4 MULTIPLE SUMS 39
Multiple summation has an interesting connection with the general operation
of changing the index of summation in single sums. We know by the
commutative law that
t ak = a,(k) 1
&K p(k)EK
if p(k) is any permutation of the integers. But what happens when we replace
k by f(j), where f is an arbitrary function
f: J --+ K
that takes an integer j E J into an integer f(j) E K? The general formula for
index replacement is
x Of(j) = x ak#f-(k)) (2.35)
jCJ
kEK
where #f-(k) stands for the number of elements in the set
f-(k) = {j If(j) = k> y
that is, the number of values of j E J such that f(j) equals k.
It’s easy to prove (2.35) by interchanging the order of summation,
x (h(j)
= x ak [f(j)=k] = x akt[f(j)=k] ,
jEJ jEJ kEK jCJ
&K
since xjEJ[f(j) =k] = #f-(k). In the special case that f is a one-to-one
My other math correspondence between J and K, we have #f-(k) = 1 for all k, and the
teacher calls this a
“bijection”; maybe
general formula (2.35) reduces to
171 learn to love
that word some day. x af(j) = t af(j) = xak.
And then again. . .
jEJ f(jlEK kEK
This is the commutative law (2.17) we had before, slightly disguised.
Our examples of multiple sums so far have all involved general terms like
ok or bk. But this book is supposed to be concrete, so let’s take a look at a
multiple sum that involves actual numbers: