The number of ways a Set of elements can be Partitioned into nonempty Subsets is called a Bell Number and is denoted . For example, there are five ways the numbers 1, 2, 3 can be partitioned: , , , , and , so . and the first few Bell numbers for , 2, ... are 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, ... (Sloane's A000110). Bell numbers are closely related to Catalan Numbers.

The diagram below shows the constructions giving and , with line segments representing elements in the same Subset and dots representing subsets containing a single element (Dickau).

The Integers can be defined by the sum

(1) |

(2) |

(3) |

(4) |

The Bell number is also equal to , where is a Bell Polynomial. Dobinski's
Formula gives the th Bell number

(5) |

(6) |

(7) |

(8) |

(9) |

Touchard's Congruence states

(10) |

(11) |

**References**

Bell, E. T. ``Exponential Numbers.'' *Amer. Math. Monthly* **41**, 411-419, 1934.

Comtet, L. *Advanced Combinatorics.* Dordrecht, Netherlands: Reidel, 1974.

Conway, J. H. and Guy, R. K. In *The Book of Numbers.* New York: Springer-Verlag, pp. 91-94, 1996.

de Bruijn, N. G. *Asymptotic Methods in Analysis.* New York: Dover, pp. 102-109, 1958.

Dickau, R. M. ``Bell Number Diagrams.'' http://forum.swarthmore.edu/advanced/robertd/bell.html.

Gardner, M. ``The Tinkly Temple Bells.'' Ch. 2 in
*Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine.*
New York: W. H. Freeman, 1992.

Gould, H. W. *Bell & Catalan Numbers: Research Bibliography of Two Special Number Sequences, 6th ed.*
Morgantown, WV: Math Monongliae, 1985.

Lenard, A. In
*Fractal Music, Hypercards, and More Mathematical Recreations from Scientific American Magazine.* (M. Gardner).
New York: W. H. Freeman, pp. 35-36, 1992.

Levine, J. and Dalton, R. E. ``Minimum Periods, Modulo , of First Order Bell Exponential Integrals.''
*Math. Comput.* **16**, 416-423, 1962.

Lovász, L. *Combinatorial Problems and Exercises, 2nd ed.* Amsterdam, Netherlands:
North-Holland, 1993.

Pitman, J. ``Some Probabilistic Aspects of Set Partitions.'' *Amer. Math. Monthly* **104**, 201-209, 1997.

Rota, G.-C. ``The Number of Partitions of a Set.'' *Amer. Math. Monthly* **71**, 498-504, 1964.

Sloane, N. J. A. Sequence
A000110/M1484
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

© 1996-9

1999-05-26