
In Many Circles. Permutations as Products of Cycles. 115
THEOREM 3.49
Let K be a field of characteristic zero, and let f
i
: N → K be functions,
1 ≤ i ≤ k. Define h : N → K by
h(n)=
f
1
(|A
1
|)f
2
(|A
2
|) ···f
k
(|A
k
|),
where the sum ranges over all weak ordered partitions (A
1
,A
2
, ···,A
k
) of [n]
into k parts. Let F
i
(x) and H(x) be the exponential generating functions of
the sequences f
i
(n) and h(n). Then we have
H(x)=F
1
(x)F
2
(x) ···F
n
(x).
You could ask what the advantage of letting f and h map into any field of
characteristic zero is, as opposed to just the field of real or complex numbers.
The answer will become obvious when we discuss cycle indices later in this
section. ...