Concrete mathematics : a foundation for computer science

Table 253 Stirling’s triangles in tandem.
6.1 STIRLING NUMBERS 253
n {:5} {_nq} {:3} {:2} {:1} {i} {Y} {I} {3} { a } {r}
-5 1
-4 10 1
-3 35 6 1
-2 50 11 3 1
-1 24 6 2 1 1
0 0 0 0 0 0 1
1 0 0 0 0 0 0 1
2 0 0 0 0 0 0 11
3 0 0 0 0 0 0 13 1
4 0 0 0 0 0 0 17 6 1
5 0 0 0 0 0 0 115 25 10 1
In fact, a surprisingly pretty pattern emerges: Stirling’s triangle for cycles
appears above Stirling’s triangle for subsets, and vice versa! The two kinds
of Stirling numbers are related by an extremely simple law:
[I] = {I:}, integers k,n.
We have “duality,” something like the relations between min and max, between
1x1 and [xl, between XL and xK, between gcd and lcm. It’s easy to check that
both of the recurrences [J = (n- 1) [“;‘I + [i;:] and {i} = k{n;‘} + {:I:}
amount to the same thing, under this correspondence.
6.2 EULERIAN NUMBERS
Another triangle of values pops up now and again, this one due to
Euler [88, page 4851, and we denote its elements by (E). The angle brackets
in this case suggest “less than” and “greater than” signs; (E) is the number of
permutations rr1 rr2 . . . rr, of {l ,2, . . . , n} that have k ascents, namely, k places
where Xj < nj+l. (Caution: This notation is even less standard than our notations
[t] , {i} for Stirling numbers. But we’ll see that it makes good sense.)
For example, eleven permutations of {l ,2,3,4} have two ascents:
1324, 1423, 2314, 2413, 3412;
1243, 1342, 2341; 2134, 3124, 4123.
(The first row lists the permutations with ~1 < 7~2 > 7r3 < 7~; the second row
lists those with rrl < ~2 < 7~3 > 7~4 and ~1 > rr2 < 713 < 7r4.) Hence (42) = 11.