Bell numbers

2020 Mathematics Subject Classification: Primary: 11B73 [MSN][ZBL]

The Bell numbers $B_0,B_1,\ldots$ are given by

$$\sum_{n=0}^\infty B_n\frac{x^n}{n!}=e^{e^x-1}$$

or by

$$B_{n+1}=\sum_{k=0}^n\binom nkB_k.$$

Also,

$$B_n=\sum_{k=1}^nS(n,k),$$

where $S(n,k)$ are Stirling numbers of the second kind (cf. Combinatorial analysis), so that $B_n$ is the total number of partitions of an $n$-set.

They are equal to $1,1,2,5,15,52,203,877,4140,\ldots$ (OEIS sequence A000110).

The name honours E.T. Bell.

References

• L. Comtet, "Advanced combinatorics", Reidel (1974) Zbl 0283.05001