CALCULATE WITH CONFIDENCE

Counting, Permutations & Combinations

Bell Number Calculator

Count all partitions of n labeled objects into any number of non-empty unlabeled groups.

Counting & Combinatorics

Count every partition of a labeled set

Exact Bell number
  1. 1EnterProvide the known values
  2. 2CalculateResults update automatically
  3. 3VerifyReview the details and units
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Exact count

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Understand the subject

What does a Bell number count?

A Bell number counts every way to partition labelled objects into one or more nonempty unlabeled groups, allowing any number of groups.

Use it when the grouping structure matters but the number of groups is not fixed.

The relationship

The counting rule

See the structure

Add every allowed group count

Worked example

Check the rule with a small case

All partitions of 4 labelled objects total B(4) = 15.

Interpret with care

Choose the right counting model

A Bell number is not for labelled groups or a fixed group count.

Use the Permutation & Combination Generator when a small case needs to be inspected rather than only counted.

Quick guide

How to use this calculator

  1. Enter the number of labeled objects.
  2. All possible non-empty group counts are included.
  3. Read the exact Bell number B(n).

Calculation method

Sum the set partitions over every group count

B(n) is the sum of S(n,k) from k=0 through n.

Worked example

Group four labeled objects

Across one, two, three, or four groups, there are fifteen set partitions.

B(4) = 15

Supported inputs

Precision and limits

Discrete domain

Inputs are whole counts. Negative, fractional, grouped, and scientific-notation inputs are rejected.

Exact integers

Results use arbitrary-precision integers and are never rounded. Inputs and output size are bounded to keep the page responsive.

Maximum n

n is limited to 300 for responsive exact evaluation.