Show that the number of ordered pairs (1,2) satisfying 1 2 = * for some fixed

Show that the number of ordered pairs (ϒ1,ϒ2) satisfying ϒ1 ⋁ ϒ2 = ϒ* for some fixed partition ϒ*, is

∑

ϒ≤ϒ

∗

m (ϒ, ϒ

∗)B 2

|υ1

| …B 2

|υν

|

, where m(ϒ, ϒ*) is the Möbius function for the partition lattice, defined below

(3.12). Hence prove that the total number of ordered triplets (ϒ1,ϒ2,ϒ3) satisfying

ϒ1 ⋁ ϒ2 ⋁ ϒ3 = 1 is C

(3)

p = ∑

ϒ

(−1)

ν−1

(ν − 1)!B 3

|υ1| …B 3

|υν|

.

Show also, in the notation of Exercise 3.22, that C

(3)

p is the pth cumulant of the triple product Y = X1X2X3.