Partitioning of a Set

A partition of a set S is a collection of n disjoint subsets P₁, P₂, ... P_n that satisfies the following three conditions −

• No subset is empty − P_i ≠ ∅ for all 0 < i ≤ n
• Union covers the entire set − P₁ ∪ P₂ ∪ ... ∪ P_n = S
• Subsets are mutually disjoint − P_a ∩ P_b = ∅ for a ≠ b

Example

Let S = { a, b, c, d, e, f, g, h }. Some valid partitions are −

Partition 1:  { a }, { b, c, d }, { e, f, g, h }
Partition 2:  { a, b }, { c, d }, { e, f, g, h }
Partition 3:  { a, b, c, d, e, f, g, h }           (single subset)
Partition 4:  { a }, { b }, { c }, { d }, { e }, { f }, { g }, { h }  (all singletons)

Each partition satisfies all three conditions − no empty subsets, their union equals S, and no two subsets overlap.

Bell Numbers

Bell numbers give the count of the total number of ways to partition a set. They are denoted by B_n where n is the cardinality of the set.

Example

Let S = { 1, 2, 3 }, n = |S| = 3. All possible partitions are −

1.  { 1, 2, 3 }                 (one group of all)
2.  { 1 }, { 2, 3 }             (1 separated)
3.  { 1, 2 }, { 3 }             (3 separated)
4.  { 1, 3 }, { 2 }             (2 separated)
5.  { 1 }, { 2 }, { 3 }         (all separated)

Hence B₃ = 5.

The first few Bell numbers are −

Conclusion

A partition divides a set into non-empty, non-overlapping subsets whose union reconstructs the original set. The Bell number B_n counts the total number of distinct partitions possible for a set of n elements.