Concrete mathematics : a foundation for computer science

Table 245 Stirling’s triangle for cycles.
n
0
1
2c3
4
5
6
7
8
9
1
0
0
0
0
0
0
0
0
0
1
1
2
6
24
120
720
5040
40320
1
3 1
6.1 STIRLING NUMBERS 245
11 6 1
50 35 10 1
274 225 85 15 1
1764 1624 735 175 21 1
13068 13132 6769 1960 322 28 1
109584 118124 67284 22449 4536 546 36 1
A modification of this argument leads to a recurrence by which we can
compute {L} for all k: Given a set of n > 0 objects to be partitioned into k
nonempty parts, we either put the last object into a class by itself (in {:I:}
ways), or we put it together with some nonempty subset of the first n - 1
objects. There are k{n,‘} possibilities in the latter case, because each of the
{ “;‘} ways to distribute the first n - 1 objects into k nonempty parts gives
k subsets that the nth object can join. Hence
{;1) = k{rrk’}+{EI:}, integern>O.
This is the law that generates Table 244; without the factor of k it would
reduce to the addition formula (5.8) that generates Pascal’s triangle.
And now, Stirling numbers of the first kind. These are somewhat like
the others, but [L] counts the number of ways to arrange n objects into k
cycles instead of subsets. We verbalize ‘[;I’ by saying “n cycle k!’
Cycles are cyclic arrangements, like the necklaces we considered in Chapter
4. The cycle
can be written more compactly as ‘[A, B, C, D]‘, with the understanding that
[A,B,C,D] = [B,C,D,A] = [C,D,A,Bl = [D,A,B,Cl;
a cycle “wraps around” because its end is joined to its beginning. On the other
hand, the cycle [A, B, C, D] is not the same as [A, B, D, C] or [D, C, B, A].