Concrete mathematics : a foundation for computer science

Znvert this:
‘zmb ppo’.
5.3 TRICKS OF THE TRADE 193
This dual relationship between f and g is called an inversion formula; it’s
rather like the Mobius inversion formulas (4.56) and (4.61) that we encountered
in Chapter 4. Inversion formulas tell us how to solve “implicit recurrences,”
where an unknown sequence is embedded in a sum.
For example, g(n) might be a known function, and f(n) might be unknown;andwemighthavefoundawaytoprovethatg(n)
=tk(t)(-l)kf(k).
Then (5.48) lets us express f(n) as a sum of known values.
We can prove (5.48) directly by using the basic methods at the beginning
of this chapter. If g(n) = tk (T)(-l)kf(k) for all n 3 0, then
x (3 (-1 )kg(k) = F (3 t-1 lk t (r) C-1 )‘f(i)
k i
= tfiii; (11)1-ilk+‘(F)
i
= xfij)& G)(-llk+‘(~?)
i
= ~f(i,(~) F(-l)*(nij)
i
[n-j=01 = f(n).
The proof in the other direction is, of course, the same, because the relation
between f and g is symmetric.
Let’s illustrate (5.48) by applying it to the “football victory problem”:
A group of n fans of the winning football team throw their hats high into the
air. The hats come back randomly, one hat to each of the n fans. How many
ways h(n, k) are there for exactly k fans to get their own hats back?
For example, if n = 4 and if the hats and fans are named A, B, C, D,
the 4! = 24 possible ways for hats to land generate the following numbers of
rightful owners:
ABCD 4 BACD 2 CABD 1 DABC 0
ABDC 2 BADC 0 CADB 0 DACB 1
ACBD 2 BCAD 1 CBAD 2 DBAC 1
ACDB 1 BCDA 0 CBDA 1 DBCA 2
ADBC 1 BDAC 0 CDAB 0 DCAB 0
ADCB 2 BDCA 1 CDBA 0 DCBA 0
Therefore h(4,4) = 1; h(4,3) = 0; h(4,2) = 6; h(4,l) = 8; h(4,O) = 9.