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Bell Number

Bell Number
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The number of ways a set of n elements can be partitioned into nonempty subsets is called a Bell number and is denoted B_n (not to be confused with the Bernoulli number, which is also commonly denoted B_n).

For example, there are five ways the numbers {1,2,3} can be partitioned: {{1},{2},{3}}{{1,2},{3}}{{1,3},{2}}{{1},{2,3}}, and {{1,2,3}}, so B_3=5. (The explicit set partitions on {1,2,...,n} can be enumerated using SetPartitions[n] in theMathematica package Combinatorica` .)

B_0=1, and the first few Bell numbers for n=1, 2, ... are 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, ... (Sloane's A000110). The numbers of digits in B_(10^n) for n=0, 1, ... are given by 1, 6, 116, 1928, 27665, ... (Sloane's A113015).

Bell numbers are implemented in Mathematica as BellB[n].

Though Bell numbers have traditionally been attributed to E. T. Bell as a result of the general theory he developed in his 1934 paper (Bell 1934), the first systematic study of Bell numbers was made by Ramanujan in chapter 3 of his second notebook approximately 25-30 years prior to Bell's work (B. C. Berndt, pers. comm., Jan. 4 and 13, 2010).

The first few prime Bell numbers occur at indices n=2, 3, 7, 13, 42, 55, 2841, ... (Sloane's A051130), with no others less than 30447 (Weisstein, Apr. 23, 2006). These correspond to the numbers 2, 5, 877, 27644437, ... (Sloane's A051131). B_(2841) was proved prime by I. Larrosa Canestro in 2004 after 17 months of computation using the elliptic curve primality proving program PRIMO.


Bell numbers are closely related to Catalan numbers. The diagram above shows the constructions giving B_3=5 and B_4=15, with line segments representing elements in the same subset and dots representing subsets containing a single element (Dickau). The integers B_n can be defined by the sum


where S(n,k) is a Stirling number of the second kind, i.e., as the Stirling transform of the sequence 1, 1, 1, ....

The Bell numbers are given in terms of generalized hypergeometric functions by


(K. A. Penson, pers. comm., Jan. 14, 2007).

The Bell numbers can also be generated using the sum and recurrence relation

 B_n=sum_(k=0)^(n-1)B_k(n-1; k),

where (a; b) is a binomial coefficient, using the formula of Comtet (1974)


for n>0, where [x] denotes the ceiling functionDobiński's formula gives the nth Bell number


A variation of Dobiński's formula gives

B_n = sum_(k=1)^(n)(k^n)/(k!)sum_(j=0)^(n-k)((-1)^j)/(j!)
= sum_(m=1)^(n)(m^n!(n-m))/(Gamma(m+1)Gamma(n-m+1))

where !n is a subfactorial (Pitman 1997). Another double sum is given by


The Bell numbers are given by the generating function

G(x) = 1/esum_(k=0)^(infty)1/((1-kx)k!)
= sum_(k=0)^(infty)(x^k)/((-x)^k((x-1)/x)_k)
= (_1F_1(-1/x;(x-1)/x;1))/e
= ((-1)^(1/x)[xGamma(1-x^(-1))+Gamma(-x^(-1),-1)])/(ex)
= sum_(n=0)^(infty)B_nx^n
= 1+x+2x^2+5x^3+15x^4+52x^5+...,

and the exponential generating function


An amazing integral representation for B_n was given by Cesàro (1885),

B_n = (2n!)/(pie)I[int_0^pie^(e^(e^(ntheta)))sin(ntheta)dtheta]
= (2n!)/(pie)int_0^pie^(e^(costheta)cos(sint))sin[e^(costheta)sin(sintheta)]sin(ntheta)dtheta

(Becker and Browne 1941, Callan 2005), where I[z] denotes the imaginary part of z.

The Bell number B_n is also equal to phi_n(1), where phi_n(x) is a Bell polynomial.

de Bruijn (1981) gave the asymptotic formula


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


where lambda(n) is given by


with W(n) the Lambert W-function (Graham et al. 1994, p. 493). Odlyzko (1995) gave


Touchard's congruence states

 B_(p+k)=B_k+B_(k+1) (mod p),

when p is prime. This gives as a special case for k=0 the congruence

 B_p=2 (mod p)

for n prime. It has been conjectured that

 B_(n+(p^p-1)/(p-1))=B_n (mod p)

gives the minimum period of B_n (mod p). The sequence of Bell numbers {B_1,B_2,...} is periodic (Levine and Dalton 1962, Lunnon et al. 1979) with periods for moduli m=1, 2, ... given by 1, 3, 13, 12, 781, 39, 137257, 24, 39, 2343, 28531167061, 156, ... (Sloane's A054767).

The Bell numbers also have the curious property that

|B_0 B_1 B_2 ... B_n; B_1 B_2 B_3 ... B_(n+1); | | | ... |; B_n B_(n+1) B_(n+2) ... B_(2n)| = product_(i=1)^(n)i!
= G(n+2)

(Lenard 1992), where the product is simply a superfactorial and G(n) is a Barnes G-function, the first few of which for n=0, 1, 2, ... are 1, 1, 2, 12, 288, 34560, 24883200, ... (Sloane's A000178).

SEE ALSO: Bell PolynomialBell TriangleComplementary Bell NumberDobiński's FormulaInteger Sequence PrimesStirling Number of the Second KindTouchard's Congruence


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Bell, E. T. "Exponential Numbers." Amer. Math. Monthly 41, 411-419, 1934.

Blasiak, P.; Penson, K. A.; and Solomon, A. I. "Dobiński-Type Relations and the Log-Normal Distribution." J. Phys. A: Math. Gen. 36, L273-278, 2003.

Callan, D. "Cesàro's integral formula for the Bell numbers (corrected)." Oct. 3, 2005. http://www.stat.wisc.edu/~callan/papersother/cesaro/cesar....

Cesàro, M. E. "Sur une équation aux différences mêlées." Nouv. Ann. Math. 4, 36-40, 1885.

Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. 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, 1981.

Dickau, R. M. "Bell Number Diagrams." http://mathforum.org/advanced/robertd/bell.html.

Dickau, R. "Visualizing Combinatorial Enumeration." Mathematica in Educ. Res. 8, 11-18, 1999.

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, pp. 24-38, 1992.

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

Graham, R. L.; Knuth, D. E.; and Patashnik, O. Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.

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.

Larrosa Canestro, I. "Bell(2841) Is Prime." Feb. 13, 2004. http://groups.yahoo.com/group/primenumbers/message/14558.

Levine, J. and Dalton, R. E. "Minimum Periods, Modulo p, 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.

Lunnon, W. F.; Pleasants, P. A. B.; and Stephens, N. M. "Arithmetic Properties of Bell Numbers to a Composite Modulus, I." Acta Arith. 35, 1-16, 1979.

Odlyzko, A. M. "Asymptotic Enumeration Methods." In Handbook of Combinatorics, Vol. 2 (Ed. R. L. Graham, M. Grötschel, and L. Lovász). Cambridge, MA: MIT Press, pp. 1063-1229, 1995. http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.pdf.

Penson, K. A.; Blasiak, P.; Duchamp, G.; Horzela, A.; and Solomon, A. I. "Hierarchical Dobiński-Type Relations via Substitution and the Moment Problem." 26 Dec 2003. http://www.arxiv.org/abs/quant-ph/0312202/.

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.

Sixdeniers, J.-M.; Penson, K. A.; and Solomon, A. I. "Extended Bell and Stirling Numbers from Hypergeometric Functions." J. Integer Sequences 4, No. 01.1.4, 2001. http://www.math.uwaterloo.ca/JIS/VOL4/SIXDENIERS/bell.html.

Sloane, N. J. A. Sequences A000110/M1484, A000178/M2049, A051130A051131A054767, and A113015 in "The On-Line Encyclopedia of Integer Sequences."

Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cambridge, England: Cambridge University Press, pp. 33-34, 1999.

Stanley, R. P. Enumerative Combinatorics, Vol. 2. Cambridge, England: Cambridge University Press, p. 13, 1999.

Wilson, D. "Bell Number Question." math-fun@cs.arizona.edu mailing list. 16 Jul 2007.


Weisstein, Eric W. "Bell Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BellNumber.html


Source : http://mathworld.wolfram.com/BellNumber.html

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