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A008277
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Triangle of Stirling numbers of 2nd kind, S2(n,k), n >= 1, 1<=k<=n.
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1, 1, 1, 1, 3, 1, 1, 7, 6, 1, 1, 15, 25, 10, 1, 1, 31, 90, 65, 15, 1, 1, 63, 301, 350, 140, 21, 1, 1, 127, 966, 1701, 1050, 266, 28, 1, 1, 255, 3025, 7770, 6951, 2646, 462, 36, 1, 1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Also known as Stirling set numbers and written {n, k}. S2(n,k) enumerates partitions of an n-set into k non-empty subsets.
Triangle S2(n,k), 1<=k<=n, read by rows, given by [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, 6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...] where DELTA is Deleham's operator defined in A084938.
Number of partitions of {1, ...,n+1} into k+1 subsets of nonconsecutive integers, including the partition 1|2|...|n+1 if n=k. E.g. S2(3,2)=3 since the number of partitions of {1,2,3,4} into three subsets of nonconsecutive integers is 3, i.e., 13|2|4, 14|2|3, 1|24|3. - A. O. Munagi (amunagi(AT)yahoo.com), Mar 20 2005
Draw n cards (with replacement) from a deck of k cards. Let prob(n,k) be the probability that each card was drawn at least once. Then prob(n,k) = S2(n,k)*k!/k^n (see A090582). - Rainer Rosenthal (r.rosenthal(AT)web.de), Oct 22 2005
Define f_1(x),f_2(x),... such that f_1(x)=e^x and for n=2,3,... f_{n+1}(x)=diff(x*f_n(x),x). Then f_n(x)=e^x*sum(stirling2(n,k)*x^(k-1),k=1..n). - Milan R. Janjic (agnus(AT)blic.net), May 30 2008
Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 03 2008: (Start)
For tables of restricted Stirling numbers of the second kind see A143494 - A143496.
Stirling2(n,k) gives the number of 'patterns' of words of length n using k distinct symbols - see [Cooper & Kennedy] for an exact definition of the term 'pattern'. As an example, the words AADCBB and XXEGTT, both of length 6, have the same pattern of letters. The five patterns of words of length 3 are AAA, AAB, ABA, BAA and ABC giving row 3 of this table as (1,3,1).
Equivalently, Stirling2(n,k) gives the number of sequences of positive integers (N_1,...,N_n) of length n, with k distinct entries, such that N_1 = 1 and N_(i+1) <= 1 + max{j = 1..i} N_j for i >= 1. For example, Stirling(4,2) = 7 since the sequences of length 4 having 2 distinct entries that satisfy the conditions are (1,1,1,2), (1,1,2,1), (1,2,1,1), (1,1,2,2), (1,2,2,2), (1,2,2,1) and (1,2,1,2). (End)
Number of combinations of subsets in the plane. [From Mats Granvik (mats.granvik(AT)abo.fi), Jan 13 2009]
Contribution from Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Apr 06 2009: (Start)
S2(n+1,k+1) is the number of size k collections of pairwise disjoint, nonempty subsets of [n]. For example: S2(4,3)=6 because there are six such collections of subsets of [3] that have cardinality two: {(1)(23)},{(12)(3)},{(13)(2)},
{(1)(2)},{(1)(3)},{(2)(3)} (End)
Consider a set of A000217(n) balls of n colors in which, for each integer k = 1 to n, exactly one color appears in the set a total of k times. (Each ball has exactly one color and is indistinguishable from other balls of the same color.) a(n+1, k+1) equals the number of ways to choose 0 or more balls of each color in such a way that exactly (n-k) colors are chosen at least once, and no two colors are chosen the same positive number of times. [From Matthew Vandermast(ghodges14(AT)comcast.net), Nov 22 2010]
S2(n,k) is the number of monotonic-labeled forests on n vertices with exactly k rooted trees, each of height one or less. See link "Counting forests with Stirling and Bell numbers" below. [From Dennis P. Walsh, Nov 16 2011]
If D is the operator d/dx, and E the operator xd/dx, Stirling numbers are given by: E^n = sum_{k=1}^n s2(n,k) * x^k*D^k. - Hyunwoo Jang, Dec 13 2011
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REFERENCES
| M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
Tewodros Amdeberhan, Valerio de Angelis and Victor H. Moll, Complementary Bell numbers: arithmetical properties and Wilf's conjecture, http://129.81.170.14/~vhm/papers_html/final-bell.pdf.
Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, Arxiv preprint arXiv:1105.3044, 2011
A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 103ff.
Moussa Benoumhani, The Number of Topologies on a Finite Set, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
Bleick, W. E. and Wang, Peter C. C., Asymptotics of Stirling numbers of the second kind. Proc. Amer. Math. Soc. 42 (1974), 575-580.
Bleick, W. E. and Wang, Peter C. C., Erratum to: "Asymptotics of Stirling numbers of the second kind" (Proc. Amer. Math. Soc. {42} (1974), 575-580). Proc. Amer. Math. Soc. 48 (1975), 518.
B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8. English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 42.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 310.
J. H. Conway and R. K. Guy, The Book of Numbers, Springer, p. 92.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.7.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 244.
M. Griffiths, Remodified Bessel Functions via Coincidences and Near Coincidences, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
Hsu, L. C., Note on an asymptotic expansion of the nth difference of zero. Ann. Math. Statistics 19 (1948), 273-277.
Yoshinari Inaba, Hyper-Sums of Powers of Integers and the Akiyama-Tanigawa Matrix, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.7.
Knessl, Charles and Keller, Joseph B., Stirling number asymptotics from recursion equations using the ray method. Stud. Appl. Math. 84 (1991), no. 1, 43-56.
Korshunov, A. D., Asymptotic behavior of Stirling numbers of the second kind. (Russian) Metody Diskret. Analiz. No. 39 (1983), 24-41.
Nate Kube and Frank Ruskey, Sequences That Satisfy a(n-a(n))=0, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
E. Kuz'min and A. I. Shirshov: On the number e, pp. 111-119, eq.(6), in: Kvant Selecta: Algebra and Analysis, I, ed. S. Tabachnikov, Am.Math.Soc., 1999, p. 116, eq. (11).
N. Moreira and R. Reis, On the Density of Languages Representing Finite Set Partitions, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
A. O. Munagi, k-Complementing Subsets of Nonnegative Integers, International Journal of Mathematics and Mathematical Sciences, 2005:2, (2005), 215-224.
J. Riordan, An Introduction to Combinatorial Analysis, p. 48.
J. Stirling, The Differential Method, London, 1749; see p. 7.
Temme, N. M. Asymptotic estimates of Stirling numbers. Stud. Appl. Math. 89 (1993), no. 3, 233-243.
Timashev, A. N. On asymptotic expansions of Stirling numbers of the first and second kinds. (Russian) Diskret. Mat. 10 (1998), no. 3,148-159 translation in Discrete Math. Appl. 8 (1998), no. 5, 533-544.
M. C. Wolf, Symmetric functions for non-commutative elements, Duke Math. J., 2 (1936), 626-637.
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LINKS
| T. D. Noe, First 100 rows, flattened
Joerg Arndt, Fxtbook, pp.358-360
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem.
C. Cooper and R. E. Kennedy, Patterns, automata and Stirling numbers of the second kind, Mathematics and Computer Education Journal, 26 (1992), 120-124. [From Peter Bala (pbala(AT)toucansurf.com), Oct 03 2008]
T. Copeland, The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions
T. Copeland, Mathemagical Forests
T. Copeland, Addendum to Mathemagical Forests
R. M. Dickau, Stirling numbers of the second kind
G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, One-parameter groups and combinatorial physics.
W. Lang, On generalizations of Stirling number triangles, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
A. O. Munagi, k- Complementing Subsets of Nonnegative Integers, International Journal of Mathematics and Mathematical Sciences, 2005:2 (2005), 215-224.
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem
K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp and A. I. Solomon, Laguerre-type derivatives: Dobinski relations and combinatorial identities, J. Math. Phys. vol. 50, 083512 (2009)
S. Ramanujan, Notebook entry
A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela and K. A. Penson, Partition functions and graphs: A combinatorial approach.
Dennis Walsh, Counting forests with Stirling and Bell numbers
Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind
Eric Weisstein's World of Mathematics, Stirling numbers of the 2nd kind.
Eric Weisstein's World of Mathematics, Differential Operator
Thomas Wieder, The number of certain k-combinations of an n-set, Applied Mathematics Electronic Notes, vol. 8 (2008).
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, pp. 17ff, 105ff.
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FORMULA
| S2(n, k) = k*S2(n-1, k)+S2(n-1, k-1), n>1. S2(1, k) = 0, k>1. S2(1, 1)=1.
E.g.f.: A(x, y)=exp(y*exp(x)-y). E.g.f. for m-th column: ((exp(x)-1)^m)/m!.
S2(n, k) = (1/k!) * Sum_{i=0..k} (-1)^(k-i)*C(k, i)*i^n.
Row sums: Bell number A000110(n) = sum(S2(n, k)) k=1..n, n>0.
A019538(n, k)=k!* S2(n, k).
A028248(n, k) = (k-1)!*S2(n, k).
For asymptotics see Hsu (1948), among other sources.
The k-th row (k>=1) contains a(n, k) for n=1 to k where a(n, k) = (1/(n-1)!) * Sum_{q=1..[2*n+1+(-1)^(n-1)]/4} [ C(n-1, 2*q-2)*(n-2*q+2)^(k-1) -C(n-1, 2*q-1)*(n-2*q+1)^(k-1) ]. E.g. Row 7 contains S2(7, 3)=301 { A001298, S2(n+4, n) } and will be computed as the following: S2(7, 3) = a(3, 7) = 1/(3-1)! * Sum_{q=1..2} [ C(3-1, 2*q-2)*(3-2*q+2)^(7-1) -C(3-1, 2*q-1)*(3-2*q+1)^(7-1) ] = Sum_{q=1..2} [ C(2, 2*q-2)*(5-2*q)^6 -C(2, 2*q-1)*(4-2*q)^6 ]/2! = [ C(2, 0)*3^6 -C(2, 1)*2^6 +C(2, 2)*1^6 -C(2, 3)*0^6 ]/2! = [ 729 -128 +1 -0 ]/2 = 301. - Andre F. Labossiere (boronali(AT)laposte.net), Jun 07 2004
For k>0 and for all x sufficiently small, Sum[n>=0, T(n, k)*x^n ] = x^k/[(1-x)(1-2x)(1-3x)...(1-kx)].
With P(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j, i) = the j-th part of the i-th partition of n, m(i, j) = multiplicity of the j-th part of the i-th partition of n, sum_[p(i)=m]_{i=1}^{P(n)} = sum running from i=1 to i=p(n) but taking only partitions with p(i)=m parts into account, prod_{j=1}^{p(i)} = product running from j=1 to j=p(i), prod_{j=1}^{d(i)} = product running from j=1 to j=d(i) one has S2(n, m) = sum_[p(i)=m]_{i=1}^{P(n)} (n!/prod_{j=1}^{p(i)} p(i, j)!) (1/prod_{j=1}^{d(i)} m(i, j)!). For example, S2(6, 3) = 90 because n=6 has the following partitions with m=3 parts: (114), (123), (222). Their complexions are: (114): (6!/1!*1!*4!)*(1/2!*1!) = 15, (123): (6!/1!*2!*3!)*(1/1!*1!*1!) = 60, (222): (6!/2!*2!*2!)*(1/3!) = 15. The sum of the complexions is 15+60+15=90=S2(6, 3). - Thomas Wieder (wieder.thomas(AT)t-online.de), Jun 02 2005
Sum(k*S2(n,k), k=1..n)=B(n+1)-B(n), where B(q) are the Bell numbers (A000110). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 01 2006
Recurrence: S2(n+1,k) = sum_{i=0}^n C(n,i) S2(i, k-1). With the starting conditions S2(n,k) = 1 for n = 0 or k = 1 and S2(n,k) = 0 for k = 0 we have the closely related recurrence S2(n,k) = sum_{i=k}^n C(n-1,i-1) S2(i-1, k-1) where C(n,m) is the binomial coefficient. - Thomas Wieder (thomas.wieder(AT)t-online.de), Jan 27 2007
Representation of Stirling numbers of the second kind S2(n,k), n=1,2..., k=1,2...n, as special values of hypergeometric function of type (n)F(n-1): stirling2(n,k)= (-1)^(k-1)*hypergeom([ -k+1,2,2...2],[1,1...1],1)/(k-1)!, i.e. having n parameters in the numerator: one equal to -k+1 and n-1 parameters all equal to 2 ; and having n-1 parameters in the denominator all equal to 1 and the value of the argument equal to 1. Example: stirling2(6,k)= seq(evalf((-1)^(k-1)*hypergeom([ -k+1,2,2,2,2,2],[1,1,1,1,1],1)/(k-1)!),k=1..6)=1,31,90,65,15,1. - Karol A. Penson ( penson(AT)lptl.jussieu.fr ), Mar 28 2007
Formulae and comments from Tom Copeland, Oct 10 2007 (Start): Bell(n,x) = sum(j=0,...,n) S2(n,j) * x^j = sum(j=0,...,n) E(n,j) * Lag(n,-x, j-n) = sum(j=0,...,n) [ E(n,j)/n! ] * [ n!*Lag(n,-x, j-n) ] = sum(j=0,...,n) E(n,j) * C(Bell(.,x)+j,n) umbrally where Bell(n,x) are the Bell / Touchard / exponential polynomials; S2(n,j), the Stirling numbers of the second kind; E(n,j), the Eulerian numbers; Lag(n,x,m), the associated Laguerre polynomials of order m; and C(x,y) = x!/[ y!*(x-y)! ].
By substituting the umbral compositional inverse of the Bell polynomials, the lower factorial n!*C(x,n), for x in the equation, the equation becomes x^n = sum(j=0,...,n) S2(n,j) * j! * C(x,j)
Note that E(n,j)/n! = E(n,j)/ {sum(k=0,..,n)E(n,k)}. Also [n!*Lag(n,-1, j-n)] is A086885 with a simple combinatorial interpretation in terms of seating arrangements, giving a combinatorial interpretation to the equation for x = 1; n!*Bell(n,1) = n!*sum(j=0,...,n) S2(n,j) = sum(j=0,...,n) {E(n,j) * [n!*Lag(n,-1, j-n)]} . (End)
n-th row = leftmost column of nonzero terms of A127701^(n-1). Also, (n+1)-th row of the triangle = A127701 * n-th row; deleting the zeros. Example: A127701 * [1, 3, 1, 0, 0, 0,...] = [1, 7, 6, 1, 0, 0, 0,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 21 2007
Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 11 2009: (Start)
p(x,n)=Sum[m^n*x^m/m!, {m, 0, Infinity}]/(x*Exp[x]);
s(n,m)=coefficients(p(x,n)) (End)
The row polynomials are given by D^n(exp(x*t)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A147315 and A094198. See also A185422. - Peter Bala, Nov 25 2011
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EXAMPLE
| Triangle begins:
1;
1, 1;
1, 3, 1;
1, 7, 6, 1;
1, 15, 25, 10, 1;
1, 31, 90, 65, 15, 1;
1, 63, 301, 350, 140, 21, 1;
1, 127, 966, 1701, 1050, 266, 28, 1;
1, 255, 3025, 7770, 6951, 2646, 462, 36, 1;
1, 511, 9330, 34105, 42525, 22827, 5880, 750, 45, 1;
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MAPLE
| seq(seq(combinat[stirling2](n, k), k=1..n), n=1..10); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 02 2007
stirling_2 := (n, k) -> (1/k!) * add((-1)^(k-i)*binomial(k, i)*i^n, i=0..k);
Stirling2rec := proc(n::integer, k::integer) # Calculate the Stirling numbers of the second kind S2(n, k) by a recurrence. local i, Result; if k > n or n < 0 or k < 0 then return fail end if; if n = 0 or k = 1 then Result := 1; return Result end if; if k = 0 then Result := 0; return Result end if; Result := 0; for i from k to n do Result := Result + binomial(n - 1, i - 1) * Stirling2rec(i - 1, k - 1); end do; return Result; end proc; - Thomas Wieder (thomas.wieder(AT)t-online.de), Jan 27 2007
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MATHEMATICA
| Table[StirlingS2[n, k], {n, 11}, {k, n}] // Flatten (from Robert G. Wilson v (rgwv(at)rgwv.com), May 23 2006)
Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 11 2009: (Start)
p[x_, n_] = Sum[m^n*x^m/m!, {m, 0, Infinity}]/(x*Exp[x]);
Table[FullSimplify[ExpandAll[p[x, n]]], {n, 1, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 1, 10}];
Flatten[%] (End)
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PROG
| (PARI) S2(n, k) = if(k<1|k>n, 0, if(n==1, 1, k*S2(n-1, k)+S2(n-1, k-1))); printp(matrix(9, 9, n, k, S2(n, k)))
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CROSSREFS
| Cf. A008275 (Stirling numbers of first kind), A039810-A039813, A048993 (another version of this triangle), A048994, A028246.
Cf. A094262, A000217, A001296, A001297, A001298, A087127, A087107-A087111.
Cf. A127701.
Sequence in context: A130749 A154959 * A080417 A193387 A185982 A133800
Adjacent sequences: A008274 A008275 A008276 * A008278 A008279 A008280
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KEYWORD
| nonn,tabl,nice,core
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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