Recurrence for Bell numbers

Builds on Binomial coefficient Bell numbers

Statement

Let $n$ be a natural number, and let $B_m$ be the Bell numbers. Then $B_{n+1} = \sum_{k=0}^{n} \binom{n}{k} B_k.$

Remarks

Proof idea: in a partition of $\{1, \dots, n + 1\}$ , the block of $n + 1$ contains $n - k$ other elements, for some $k$ . The remaining $k$ elements form an arbitrary partition. Example: $B_4 = 1 \cdot 1 + 3 \cdot 1 + 3 \cdot 2 + 1 \cdot 5 = 15$ .

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Binomial coefficient definition
Stirling numbers of the second kind definition
Bell numbers definition
Recurrence for Bell numbers statement

Tag	`0110GBHPPB`
URI	`m2://0110GBHPPB`
Kind	statement, theorem
Writing	W0 stub
MSC	11B73
Status	This copy of the site has no build data. Run `m2 status 0110GBHPPB`.
Uses	Binomial coefficient, Bell numbers

Recurrence for Bell numbers

Statement

Remarks

Reading path

Further reading

What this page needs for W1

What links here