Difference between revisions of "Bell numbers"
From Encyclopedia of Mathematics
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| − | + | {{TEX|done}}{{MSC|11B73}} | |
| − | + | 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 | or by | ||
| − | + | $$B_{n+1}=\sum_{k=0}^n\binom nkB_k.$$ | |
Also, | Also, | ||
| − | + | $$B_n=\sum_{k=1}^nS(n,k),$$ | |
| − | where | + | 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 | + | They are equal to $1,1,2,5,15,52,203,877,4140,\ldots$ ({{OEIS|A000110}}). |
The name honours E.T. Bell. | The name honours E.T. Bell. | ||
====References==== | ====References==== | ||
| − | + | ||
| + | * L. Comtet, "Advanced combinatorics", Reidel (1974) {{ZBL|0283.05001}} | ||
Latest revision as of 07:26, 7 November 2023
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
How to Cite This Entry:
Bell numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bell_numbers&oldid=14335
Bell numbers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bell_numbers&oldid=14335
This article was adapted from an original article by N.J.A. Sloane (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article