Using the result given in Exercise 3.19, show that the total number of ordered pairs of partitions (ϒ1, ϒ2) satisfying ϒ1 ⋁ ϒ2 = 1 is C

(2)

p = ∑

ϒ

(−1)

ν−1

(ν − 1)!B 2

|υ1

| …B 2

|υν

|

where the partitions contain p elements and B 2

r is the square of the rth Bell number. Deduce also that C

(2)

p is the pth cumulant of Y = X1X2 where the Xs are

ϒ≥ϒ

∗

ϒ=(υ1,…,υν)

independent Poisson random variables with unit mean.