Discrete Mathematics
Grinshpan
The number of partitions of a set
Let X be a finite set.
A partition of X is a collection of disjoint nonempty subsets of X whose union is X.
EXAMPLE
If X = {, ∆, O}, there are 5 partitions possible:
{}, {∆}, {O}
{, ∆}, {O}
{}, {∆, O}
{, O}, {∆}
{, ∆, O}
The number of partitions is solely determined by n, the cardinality of X.
Denote this number by dn (n-th Bell’s number).
We have d1 = 1, d2 = 2, d3 = 5, d4 = 15.
PROPOSITION
For every integer n ≥ 0, dn+1
n
n
n
= d0
+ d1
+ . . . + dn
, where d0 = 1.
0
1
n
PROOF
Let X be an (n + 1)-element set, and let a be one of its elements.
Given k = 0, . . . , n, list all partitions of X that include a subset containing a and k other elements.
n
The number of such partitions is dn−k nk = dn−k n−k
.
The conclusion follows by adding over k.
An expression for dn may be given in terms of Stirling’s numbers.
If S(n, k) is the number of onto functions from an n-element set onto a k-element set,
then S(n, k)/k! gives the number of partitions of an n-element set into k nonempty subsets.
Hence, by the sum rule, dn = S(n, 1)/1! + S(n, 2)/2! + . . . + S(n, n)/n!
REMARK
z −1
It may be shown that dn is the value of the n-th derivative of ee
∞
X
5
zn
= 1 + z + z2 + z3 + . . . ,
In fact, if one sets D(z) =
dn
n!
6
n=0
z −1
then D0 (z) = ez D(z) and so D(z) = ee
.
at the origin.