Bell Number Calculator
Bell Number Calculator
Enter the number of items in a set, and this calculator finds the total number of ways to partition it into non-empty groups, using any number of groups.
A Bell number counts every possible way to partition a set regardless of how many groups end up being used — unlike a Stirling number of the second kind, which fixes the group count at a specific number k in advance. In fact, a Bell number is exactly the sum of the Stirling numbers of the second kind across every possible group count, from splitting everything into 1 single group up to putting every item in its own separate group — which is why the two calculators are closely related, and Bell numbers grow extremely quickly even for modest set sizes.
- A Bell number counts all the ways to partition a set, regardless of how many groups are used — unlike a Stirling number of the second kind, which fixes the group count at a specific k.
- The Bell number is the sum of the Stirling numbers of the second kind for every possible k, from 1 group up to n groups (where every item is its own group).
- Bell numbers grow extremely fast: 1, 1, 2, 5, 15, 52, 203, 877… — even a modest set size produces a huge number of possible partitions.
How is a Bell number different from a Stirling number of the second kind?
A Stirling number S(n,k) counts partitions into exactly k groups, while a Bell number sums S(n,k) over every possible value of k, counting partitions of any size.
How many ways can a set of 4 items be partitioned?
15 ways in total, across every possible number of groups.
Bell Number Calculator
Enter the number of items in a set, and this calculator finds the total number of ways to partition it into non-empty groups, using any number of groups.
A Bell number counts every possible way to partition a set regardless of how many groups end up being used — unlike a Stirling number of the second kind, which fixes the group count at a specific number k in advance. In fact, a Bell number is exactly the sum of the Stirling numbers of the second kind across every possible group count, from splitting everything into 1 single group up to putting every item in its own separate group — which is why the two calculators are closely related, and Bell numbers grow extremely quickly even for modest set sizes.
- A Bell number counts all the ways to partition a set, regardless of how many groups are used — unlike a Stirling number of the second kind, which fixes the group count at a specific k.
- The Bell number is the sum of the Stirling numbers of the second kind for every possible k, from 1 group up to n groups (where every item is its own group).
- Bell numbers grow extremely fast: 1, 1, 2, 5, 15, 52, 203, 877… — even a modest set size produces a huge number of possible partitions.
How is a Bell number different from a Stirling number of the second kind?
A Stirling number S(n,k) counts partitions into exactly k groups, while a Bell number sums S(n,k) over every possible value of k, counting partitions of any size.
How many ways can a set of 4 items be partitioned?
15 ways in total, across every possible number of groups.