Concrete mathematics : a foundation for computer science

246 SPECIAL NUMBERS
There are eleven different ways to make two cycles from four elements: “There are nine
and sixty ways
[1,2,31 [41, [’ ,a41 Dl , [1,3,41 PI , [&3,4 [II, of constructing
tribal lays,
[1,3,21 [41, [’ ,4,21 Dl , P,4,31 PI , P,4,31 PI,
And-every-single-
P,21 [3,41, [’ ,31 P, 4 , [I,41 P,31; one-of-them-is-
(W rjght,”
hence [“;I = 11.
-Rudyard Kipling
A singleton cycle (that is, a cycle with only one element) is essentially
the same as a singleton set (a set with only one element). Similarly, a 2-cycle
is like a 2-set, because we have [A, B] = [B, A] just as {A, B} = {B, A}. But
there are two diflerent 3-cycles, [A, B, C] and [A, C, B]. Notice, for example,
that the eleven cycle pairs in (6.4) can be obtained from the seven set pairs
in (6.1) by making two cycles from each of the 3-element sets.
In general, n!/n = (n -- 1) ! cycles can be made from any n-element set,
whenever n > 0. (There are n! permutations, and each cycle corresponds
to n of them because any one of its elements can be listed first.) Therefore
we have
n
= (n-l)!, integer n > 0.
[I 1
This is much larger than the value {;} = 1 we had for Stirling subset numbers.
In fact, it is easy to see that the cycle numbers must be at least as large as
the subset numbers,
integers n, k 3 0,
[E] 3 {L}y
because every partition into nonempty subsets leads to at least one arrangement
of cycles.
Equality holds in (6.6) when all the cycles are necessarily singletons or
doubletons, because cycles are equivalent to subsets in such cases. This happens
when k = n and when k = n - 1; hence
[Z] = {iI}’ [nl:l] = {nil}
In fact, it is easy to see that.
[“n] = {II} = ” [nil] = {nnl} = ( I )
(6.7)
(The number of ways to arrange n objects into n - 1 cycles or subsets is
the number of ways to choose the two objects that will be in the same cycle
or subset.) The triangular numbers (;) = 1, 3, 6, 10, . . . are conspicuously
present in both Table 244 and Table 245.