1,4

The unsigned numbers are also called Stirling cycle numbers: |s(n,k)| = number of permutations of n objects with exactly k cycles.

With P(n) = the number of integer partitions of n, T(i,n) = the number of parts of the i-th partition of n, D(i,n) = the number of different parts of the i-th partition of n, p(j,i,n) = the j-th part of the i-th partition of n, m(j,i,n) = multiplicity of the j-th part of the i-th partition of n, sum_[T(i,n)=k]_{i=1}^{P(n)} = sum running from i=1 to i=p(n) but taking only partitions with T(i,n)=k parts into account, prod_{j=1}^{T(i,n)} = product running from j=1 to j=T(i,n), prod_{j=1}^{D(i,n)} = product running from j=1 to j=D(i,n) one has S1(n,k) = sum_[T(i,n)=k]_{i=1}^{P(n)} (n!/prod_{j=1}^{T(i,n)} p(j,i,n))* (1/prod_{j=1}^{D(i,n)} m(j,i,n)!). For example, S1(6,3) = 225 because n=6 has the following partitions with k=3 parts: (114), (123), (222). Their complexions are: (114): (6!/1*1*4)*(1/2!*1!) = 90, (123): (6!/1*2*3)*(1/1!*1!*1!) = 120, (222): (6!/2*2*2)*(1/3!) = 15. The sum of the complexions is 90+120+15=225=S1(6,3). - Thomas Wieder (wieder.thomas(AT)t-online.de), Aug 04 2005

Row sums equal 0 - Jon Perry (perry(AT)globalnet.co.uk), Nov 14 2005

|s(n,k)| enumerates unordered n-vertex forests composed of k increasing non-plane (unordered) trees. Proof from the e.g.f. of the first column and the F. Bergeron et al. reference, especially Table 1, last row (non plane ``recursive"), given in A049029. W. Lang Oct 12 2007.

|s(n,k)| enumerates unordered increasing n-vertex k-forests composed of k unary trees (out-degree r from {0,1}) whose vertices of depth (distance from the root) j>=0 come in j+1 colors (j=0 for the k roots). W. Lang, Oct 12 2007, Feb 22 2008

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 833.

E. Barcucci, A. Del Lungo and R. Pinzani, "Deco" polyominoes, permutations and random generation, Theoretical Computer Science, 159, 1996, 29-42.

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 93ff.

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. 32.

L. Comtet, Advanced Combinatorics, Reidel, 1974; Chapter V, also p. 310.

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 93.

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 226.

H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.7.

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J. Hines, A generalization of the S-Stirling numbers, Math. Mag., 29 (1956), 200-203.

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; Keller, Joseph B. Stirling number asymptotics from recursion equations using the ray method. Stud. Appl. Math. 84 (1991), no. 1, 43-56.

B. H. Margolius, Transient and periodic solution to the time-inhomogeneous quasi-birth death process, Queueing Systems, Volume 56, Numbers 3-4 / August, 2007. [From N. J. A. Sloane, Jul 09 2009]

J. Riordan, An Introduction to Combinatorial Analysis, p. 48.

R. Sedgewick and P. Flajolet, An Introduction to the Analysis of Algorithms, Addison-Wesley, Reading, MA, 1996.

J. Stirling, The Differential Method, London, 1749; see p. 10.

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T. D. Noe, Rows 1 to 100 of triangle, flattened.

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Joerg Arndt, Fxtbook

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].

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A. F. Labossiere, Miscellaneous.

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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Thomas Wieder, Comments on A008275

s(n, k)=s(n-1, k-1)-(n-1)*s(n-1, k), n, k >= 1; s(n, 0)=s(0, k)=0; s(0, 0)=1.

The unsigned numbers a(n, k)=|s(n, k)| satisfy a(n, k)=a(n-1, k-1)+(n-1)*a(n-1, k), n, k >= 1; a(n, 0)=a(0, k)=0; a(0, 0)=1.

E.g.f. for m-th column (unsigned): ((-ln(1-x))^m)/m!.

s(n, k) = T(n-1, k-1), n>1 and k>1, where T(n, k) is the triangle [ -1, -1, -2, -2, -3, -3, -4, -4, -5, -5, -6, -6, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, 1, ...]and DELTA is Deleham's operator defined in A084938. The unsigned numbers are also |s(n, k)| = T(n-1, k-1), for n>0 and k>0, where T(n, k) = [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] DELTA [1, 0, 1, 0, 1, 0, 1, 0, ...].

Sum[(-1)^(n-i) StirlingS1[n, i] binomial[i, k], {i,0,n}] == (-1)^(n-k) StirlingS1[n+1, k+1]. - Carlo Wood (carlo(AT)alinoe.com), Feb 13 2007

G.f.: S(n) = product[j=1, n, (x-j)] (i.e. (x-1)(x-2)(x-3) = x^3 - 6x^2 + 11x - 6) - Jon Perry (perry(AT)globalnet.co.uk), Nov 14 2005

a(n,k) = s(k,n) = (-1)^(k-n) * S1(k,n) = ( (-1)^(k-n) ) * ( k!/{(n-1)!*2^(k-n)} ) * [ { 1/(k-n)! }*k^(k-n-1) - { (1/6)*(1/(k-n-2)!) }*k^(k-n-2) + { (1/72)*(1/(k-n-4)!) }*k^(k-n-3) - { (1/6480)*(5/(k-n-6)! -36/(k-n-4)!) }*k^(k-n-4) + { (1/155520)*(5/(k-n-8)!-144/(k-n-6)!) }*k^(k-n-5) - { (1/6531840)*(7/(k-n-10)! -504/(k-n-8)!+2304/(k-n-6)!) }*k^(k-n-6) + { (1/1175731200)*(35/(k-n-12)!-5040/(k-n-10)!+87264/(k-n-8)!) }*k^(k-n-7) - { (1/7054387200)*(5/(k-n-14)!-1260/(k-n-12)!+52704/(k-n-10)!-186624/(k-n-8)!) }*k^(k-n-8) + { (1/338610585600)*(5/(k-n-16)!-2016/(k-n-14)!+164736/(k-n-12)!-2156544/(k-n-10)!) }*k^(k-n-9) - ..... ]. - Andre F. Labossiere (boronali(AT)laposte.net), Mar 27 2006

|s(3,2)| = 3 for the three unordered 2-forest with 3 vertices and two increasing (non plane) trees: ((1),(2,3)), ((2),(1,3)), ((3),(1,2)).

Triangle begins:

..................................1

................................-1, 1

...............................2, -3, 1

............................-6, 11, -6, 1

.........................24, -50, 35, -10, 1

.....................-120, 274, -225, 85, -15, 1

.................720, -1764, 1624, -735, 175, -21, 1

............-5040, 13068, -13132, 6769, -1960, 322, -28, 1

......40320, -109584, 118124, -67284, 22449, -4536, 546, -36, 1

Another version of the same triangle, from Joerg Arndt, Oct 05 2009:

s(n,k) := number of permutations of n elements with exactly k cycles ("Stirling cycle numbers")

..n:...total...m=..1......2......3.....4.....5.....6....7...8...9

..1:.......1.......1

..2:.......2.......1......1

..3:.......6.......2......3......1

..4:......24.......6.....11......6.....1

..5:.....120......24.....50.....35....10.....1

..6:.....720.....120....274....225....85....15.....1

..7:....5040.....720...1764...1624...735...175....21....1

..8:...40320....5040..13068..13132..6769..1960...322...28...1

..9:..362880...40320.109584.118124.67284.22449..4536..546..36...1

|s(4,2)| = 11 for the eleven unordered 2-forest with 4 vertices, composed of two increasing (non plane) trees: ((1),((23)(24))), ((2),((13)(14)), ((3),((12)(14)), ((4),((12)(13)); ((1),(2,3,4)),((2),(1,2,3)), ((3), (1,2,4)), ((4),(1,2,3)); ((1,2),(3,4)), ((1,3),(2,4)), ((1,4),(2, 3)). - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Feb 22 2008

with (combinat):seq(seq(stirling1(n, k), k=1..n), n=1..10); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 03 2007

for i from 0 to 9 do seq(stirling1(i, j), j = 1 .. i) od; - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 29 2007

(PARI) T(n, k)=if(n<1, 0, n!*polcoeff(binomial(x, n), k))

(PARI) T(n, k)=if(n<1, 0, n!*polcoeff(polcoeff((1+x+x*O(x^n))^y, n), k))

(PARI) vecstirling(n)=Vec(factorback(vector(n-1, i, 1-i*'x))) (A function that returns all the s(n, k) as a vector) - Bill Allombert (Bill.Allombert(AT)math.u-bordeaux1.fr), Mar 16 2009

Cf. A048994, A008277 (Stirling numbers of second kind), A039814-A039817, A048993.

Cf. A084938, A094216, A008276, A094262, A008277, A008278, A121632.

A130534 is a signed version.

A000142(n) = sum(|s(n, k)|) k=0..n, n >= 0,

Sequence in context: A165675 A138771 A121748 this_sequence A130534 A107416 A105613

Adjacent sequences: A008272 A008273 A008274 this_sequence A008276 A008277 A008278

sign,tabl,nice,core

N. J. A. Sloane (njas(AT)research.att.com).