## Bell Number

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)

where is a Stirling Number of the Second Kind, or by the generating function
 (2)

The Bell numbers can also be generated using the Bell Triangle, using the Recurrence Relation
 (3)

where is a Binomial Coefficient, or using the formula of Comtet (1974)
 (4)

where denotes the Ceiling Function.

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

 (5)

Lovász (1993) showed that this formula gives the asymptotic limit
 (6)

where is defined implicitly by the equation
 (7)

A variation of Dobinski's Formula gives
 (8)

(Pitman 1997). de Bruijn (1958) gave the asymptotic formula

 (9)

Touchard's Congruence states

 (10)

when is Prime. The only Prime Bell numbers for are , , , , , and . The Bell numbers also have the curious property that
 (11)

(Lenard 1986).

See also Bell Polynomial, Bell Triangle, Dobinski's Formula, Stirling Number of the Second Kind, Touchard's Congruence

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.