Concrete mathematics : a foundation for computer science

6.7 CONTINUANTS 295
The corresponding numerator is P, = K,(zpF’, . . . , zpFn); this turns out to
be like Q,, but with fewer terms. For example, we have
compared with Q5 = 1 + z-' + .. + z--12. A closer look reveals the pattern
governing which terms are present: We have
12
p = 1 +22+z3+z5+z7+z8+z’o+z”
5
ZZ z-12
Z’2 z zm [m is F-even] ;
m=O
and in general we can prove by induction that
F,+z-’
p n = z’-Fn+~ t zm [m is F-even]
m=O
Therefore
Pll
t’,“Ji-’ z”’ [m is F-even]
-=
QTI xLL;p’ Zm ’
Taking the limit as n -+ 0;) now gives (6.146), because of (6.145).
Exercises
Warmups
1 What are the [i] = 11 permutations of {l ,2,3,4} that have exactly two
cycles? (The cyclic forms appear in (6.4); non-cyclic forms like 2314 are
desired instead.)
2 There are mn functions from a set of n elements into a set of m elements.
How many of them range over exactly k different function values?
3 Card stackers in the real world know that it’s wise to allow a bit of slack
so that the cards will not topple over when a breath of wind comes along.
Suppose the center of gravity of the top k cards is required to be at least
E units from the edge of the k + 1st card. (Thus, for example, the first
card can overhang the second by at most 1 -c units.) Can we still achieve
arbitrarily large overhang, if we have enough cards?
4 Express l/l + l/3 +... + 1/(2n+l) in terms of harmonic numbers.
5 Explain how to get the recurrence (6.75) from the definition of L&,(x, y)
in (6.74), and solve the recurrence.