Bell Number
Count partitions of an n-element labeled set as an exact decimal integer; n is capped at 300 for bounded computation.
Description
Bell numbers count all partitions of a labeled set into nonempty unlabeled blocks, regardless of block count.
Bell numbers count all partitions of a labeled set into nonempty unlabeled blocks, regardless of block count. The empty set has one partition.1
The tool returns value. The indexing and output conventions below determine how to read these values.
When to use Bell Number
Use this for the following task: Count partitions of an n-element labeled set as an exact decimal integer; n is capped at 300 for bounded computation. Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Bell Number and understand its results
- Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
- Enter n according to the field table, then run the tool.
- Read value. Preserve exact decimal strings and the stated indexing or coefficient order.
- Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| n | Set size (resource limit 300) | integer; minimum 0; maximum 300 | Required |
Build the BigInt Bell triangle, starting with [1], placing the previous row’s last value first and then adding neighboring values. Return row n’s first value.
B(n)=Σ[k=0..n]S(n,k), obtained as the first entry of Aitken triangle row n The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| value | Bell number | string (integer/arbitrary) |
{
"input": {
"n": 5
},
"output": {
"value": "52"
}
}Limitations and assumptions
The set size is at most 300. The tool returns a count without listing partitions or fixing the number of blocks.
Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.
Alternative or Complementary methods
Choose Stirling Number of the Second Kind when you need its operation: Count partitions of n labeled elements into k nonempty unlabeled blocks exactly, returning zero when k exceeds n; indices are capped at 300. Choose Bell Triangle Entry for the complementary operation: Return the zero-indexed Aitken Bell triangle entry with row zero [1], each row starting at the previous row’s last entry, and a 300-row resource limit.
Background reference: Digital Library of Mathematical Functions, §26.7.
References
-
Digital Library of Mathematical Functions, §26.7 — National Institute of Standards and Technology
Similar or alternative tools
- Stirling Number of the Second Kind
Count partitions of n labeled elements into k nonempty unlabeled blocks exactly, returning zero when k exceeds n; indices are capped at 300.
- Bell Triangle Entry
Return the zero-indexed Aitken Bell triangle entry with row zero [1], each row starting at the previous row’s last entry, and a 300-row resource limit.