Concrete mathematics : a foundation for computer science

(Attention, computer
scientists:
The stack interpretation
now applies
with respect to an
m-ary operation,
instead of the binary
multiplication
considered earlier.)
7.5 CONVOLUTIONS 347
whose total sum is +l . Such sequences can be called m-Raney sequences. If
there are k occurrences of (1 - m) and mn + 1 - k occurrences of +l , we have
k(l-m)+(mn+l-k) = 1,
hence k = n. There are (“t+‘) sequences with n occurrences of (1 - m) and
mn + 1 - n occurrences of +l, and Raney’s lemma tells us that the number
of such sequences with all partial sums positive is exactly
mn+l -
I
-
mn
=
1
( n > mn+l ( n > (m-l)n+l’
(7.66)
So this is the number of m-Raney sequences. Let’s call this a Fuss-Catalan
number Cim,“‘, because the sequence (&“‘) was first investigated by N.I.
Fuss [log] in 1791 (many years before Catalan himself got into the act). The
ordinary Catalan numbers are C, = Cr’.
Now that we k:now the answer, (7.66), let’s play “Jeopardy” and figure
out a question that leads to it. In the case m = 2 the question was: “What
numbers C, satisfy the recurrence C, = xk CkCnPiPk + (n = O]?” We will
try to find a similar question (a similar recurrence) in the general case.
The trivial sequence (+1) of length 1 is clearly an m-Raney sequence. If
we put the number (1 -m) at the right of any m sequences that are m-Raney,
we get an m-Raney sequence; the partial sums stay positive as they increase
to +2, then +3, . . . , fm, and fl . Conversely, we can show that all m-Raney
sequences (ae, . . . , ~a,,) arise in this way, if n > 0: The last term a,,,,, must
be (1 - m). The partial sums sj = a0 +. . + aj- 1 are positive for 1 < j 6 mn,
and s,, = m because s,, + a,,,,, = 1. Let kl be the largest index 6 mn such
that Sk, = 1; let k2 be largest such that skz = 2; and so on. Thus ski = j
and sk > j, for ki cc k 6 mn and 1 < j 6 m. It follows that k, = mn, and
we can verify without difficulty that each of the subsequences (ae, . . . , ok, -I),
(ok,, . . . , okJPi), . . . , (ok,,-, , . . . , ok,,, -1) is an m-Raney sequence. We must
have kl = mnl + 1, k2 - kl = mn2 + 1, . . . , k, - k,_l = mn, + 1, for
some nonnegative integers nl, n2, . . . , n,.
Therefore (“‘t’-‘) & is the answer to the following two interesting ques-
tions: “What are the numbers Cim’ defined by the recurrence
for all integers n?” “If G(z) is a power series that satisfies
G(z) = zG(z)" + 1, (7.68)
what is [z”] G(z)?”