Calculators

Bell Number Calculator

Calculate exact Bell numbers for 0 to 200 distinct items privately in your browser, with the digit count and Bell triangle method.

Count partitions of distinct items into nonempty, unlabeled groups. Exact BigInt arithmetic keeps every digit. Your data stays in your browser.

Enter a whole number from 0 to 200 using digits only. Zero items has one empty partition. Five items has 52 partitions.

Your Bell number will appear here.

Count ways to partition a set.

The Bell number B(n) counts partitions of n distinct items into nonempty, unlabeled groups. Group order and item order within a group do not matter. For three items A, B and C, the five partitions are {ABC}, {AB}{C}, {AC}{B}, {BC}{A} and {A}{B}{C}. The sequence starts B(0) = 1, B(1) = 1, B(2) = 2, B(3) = 5, B(4) = 15 and B(5) = 52. Zero items has one empty partition. This calculator builds the Bell triangle: start with the row [1]; each new row starts with the last value of the previous row, then each further entry adds the entry on its left to the entry above-left. The first value of row n is B(n). This counts all possible group counts; it does not fix the number or size of groups or handle indistinguishable items. Enter 0 to 200 using one to three digits; leading zeros and surrounding whitespace are accepted. Signs, decimals, separators and scientific notation are rejected. Exact browser BigInt arithmetic preserves every digit. Inputs stay in your browser.

Frequently Asked Questions

Everything you need to know about this tool, how it works, and privacy.

What does a Bell number count?

B(n) counts ways to partition n distinct items into nonempty groups. The groups are unlabeled, and their order does not matter. B(3) is 5 and B(5) is 52.

Why is B(0) equal to 1?

The empty set has one partition: the empty partition with no groups.

Can I specify a fixed number of groups?

No. Bell numbers count all possible numbers of groups. A fixed number of nonempty groups requires a Stirling number of the second kind.

Are results exact and private?

Yes. The calculator uses browser BigInt arithmetic for exact results from 0 to 200 items. No inputs are uploaded or stored. Copy result copies the count and explanation.