%I #590 Sep 14 2024 01:38:30
%S 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,
%T 1,1,127,966,1701,1050,266,28,1,1,255,3025,7770,6951,2646,462,36,1,1,
%U 511,9330,34105,42525,22827,5880,750,45,1,1,1023,28501,145750,246730,179487,63987,11880,1155,55,1
%N Triangle of Stirling numbers of the second kind, S2(n,k), n >= 1, 1 <= k <= n.
%C Also known as Stirling set numbers and written {n, k}.
%C S2(n,k) counts partitions of an n-set into k nonempty subsets.
%C With regard to the preceding comment: For arbitrary (including non-disjoint) covers of an n-set by k nonempty subsets see A055154. - _Manfred Boergens_, May 20 2024
%C 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 Deléham's operator defined in A084938.
%C Number of partitions of {1, ..., n+1} into k+1 nonempty 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. - _Augustine O. Munagi_, Mar 20 2005
%C 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_, Oct 22 2005
%C Define f_1(x), f_2(x), ..., such that f_1(x)=e^x and for n = 2, 3, ..., f_{n+1}(x) = (d/dx)(x*f_n(x)). Then f_n(x) = e^x*Sum_{k=1..n} S2(n,k)*x^(k-1). - _Milan Janjic_, May 30 2008
%C From _Peter Bala_, Oct 03 2008: (Start)
%C For tables of restricted Stirling numbers of the second kind see A143494 - A143496.
%C S2(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).
%C Equivalently, S2(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 (restricted growth functions). 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).
%C (End)
%C Number of combinations of subsets in the plane. - _Mats Granvik_, Jan 13 2009
%C 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)}. - _Geoffrey Critzer_, Apr 06 2009
%C 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. - _Matthew Vandermast_, Nov 22 2010
%C 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. - _Dennis P. Walsh_, Nov 16 2011
%C 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
%C The Stirling polynomials of the second kind (a.k.a. the Bell / Touchard polynomials) are the umbral compositional inverses of the falling factorials (a.k.a. the Pochhammer symbol or Stirling polynomials of the first kind), i.e., binomial(Bell(.,x),n) = x^n/n! (cf. Copeland's 2007 formulas), implying binomial(xD,n) = binomial(Bell(.,:xD:),n) = :xD:^n/n! where D = d/dx and :xD:^n = x^n*D^n. - _Tom Copeland_, Apr 17 2014
%C S2(n,k) is the number of ways to nest n matryoshkas (Russian nesting dolls) so that exactly k matryoshkas are not contained in any other matryoshka. - _Carlo Sanna_, Oct 17 2015
%C The row polynomials R(n, x) = Sum_{k=1..n} S2(n, k)*x^k appear in the numerator of the e.g.f. of n-th powers, E(n, x) = Sum_{m>=0} m^n*x^m/m!, as E(n, x) = exp(x)*x*R(n, x), for n >= 1. - _Wolfdieter Lang_, Apr 02 2017
%C With offsets 0 for n and k this is the Sheffer product matrix A007318*A048993 denoted by (exp(t), (exp(t) - 1)) with e.g.f. exp(t)*exp(x*(exp(t) - 1)). - _Wolfdieter Lang_, Jun 20 2017
%C Number of words on k+1 unlabeled letters of length n+1 with no repeated letters. - _Thomas Anton_, Mar 14 2019
%C Also coefficients of moments of Poisson distribution about the origin expressed as polynomials in lambda. [Haight] (see also A331155). - _N. J. A. Sloane_, Jan 14 2020
%C k!*S2(n,k) is the number of surjections from an n-element set to a k-element set. - _Jianing Song_, Jun 01 2022
%D 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.
%D A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 103ff.
%D B. A. Bondarenko, Generalized Pascal Triangles and Pyramids (in Russian), FAN, Tashkent, 1990, ISBN 5-648-00738-8.
%D G. Boole, Finite Differences, 5th ed. New York, NY: Chelsea, 1970.
%D C. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, 2002, Theorem 8.11, pp. 298-299.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 310.
%D J. H. Conway and R. K. Guy, The Book of Numbers, Springer, p. 92.
%D F. N. David, M. G. Kendall, and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
%D S.N. Elaydi, An Introduction to Difference Equations, 3rd ed. Springer, 2005.
%D H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.7.
%D R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 244.
%D Frank Avery Haight, Handbook of the Poisson distribution, John Wiley, 1967. See pages 6,7.
%D A. D. Korshunov, Asymptotic behavior of Stirling numbers of the second kind. (Russian) Metody Diskret. Analiz. No. 39 (1983), 24-41.
%D 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).
%D J. Riordan, An Introduction to Combinatorial Analysis, p. 48.
%D J. Stirling, The Differential Method, London, 1749; see p. 7.
%H T. D. Noe, <a href="/A008277/b008277.txt">First 100 rows, flattened</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H V. E. Adler, <a href="http://arxiv.org/abs/1510.02900">Set partitions and integrable hierarchies</a>, arXiv:1510.02900 [nlin.SI], 2015.
%H Tewodros Amdeberhan, Valerio de Angelis, and Victor H. Moll, <a href="http://www.tulane.edu/~vhm/papers_html/final-bell.pdf">Complementary Bell numbers: arithmetical properties and Wilf's conjecture</a>, Advances in Combinatorics (2013), pp. 23-56.
%H Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, pp. 358-360
%H Joerg Arndt and N. J. A. Sloane, <a href="/A278984/a278984.txt">Counting Words that are in "Standard Order"</a>
%H J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.2010">Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure</a>, arXiv:1307.2010 [math.CO], 2013-2014.
%H J. Fernando Barbero G., Jesús Salas, and Eduardo J. S. Villaseñor, <a href="http://arxiv.org/abs/1307.5624">Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications</a>, arXiv:1307.5624 [math.CO], 2013.
%H Paul Barry, <a href="http://arxiv.org/abs/1105.3044">Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays</a>, arXiv:1105.3044 [math.CO], 2011.
%H H. Belbachir, M. Rahmani, and B. Sury, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Rahmani/rahmani3.html">Sums Involving Moments of Reciprocals of Binomial Coefficients</a>, J. Int. Seq. 14 (2011) #11.6.6.
%H Hacene Belbachir and Mourad Rahmani, <a href="http://www.emis.de/journals/JIS/VOL15/Sury/sury42.html">Alternating Sums of the Reciprocals of Binomial Coefficients</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.2.8.
%H Edward A. Bender, <a href="https://doi.org/10.1016/0097-3165(73)90038-1">Central and local limit theorems applied to asymptotic enumeration</a> Journal of Combinatorial Theory, Series A, 15(1) (1973), 91-111. See Example 5.4.
%H Moussa Benoumhani, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Benoumhani/benoumhani11.html">The Number of Topologies on a Finite Set</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.6.
%H Beáta Bényi and Péter Hajnal, <a href="https://arxiv.org/abs/1804.01868">Poly-Bernoulli Numbers and Eulerian Numbers</a>, arXiv:1804.01868 [math.CO], 2018.
%H P. Blasiak, K. A. Penson, and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0212072">The Boson Normal Ordering Problem and Generalized Bell Numbers</a>, arXiv:quant-ph/0212072, 2002.
%H P. Blasiak, K. A. Penson, and A. I. Solomon, <a href="http://www.arXiv.org/abs/quant-ph/0402027">The general boson normal ordering problem</a>, arXiv:quant-ph/0402027, 2004.
%H W. E. Bleick and Peter C. C. Wang, <a href="http://dx.doi.org/10.1090/S0002-9939-1974-0330867-1">Asymptotics of Stirling numbers of the second kind</a>, Proc. Amer. Math. Soc. 42 (1974), 575-580.
%H W. E. Bleick and Peter C. C. Wang, <a href="http://dx.doi.org/10.1090/S0002-9939-1975-0401490-6">Erratum to: "Asymptotics of Stirling numbers of the second kind" (Proc. Amer. Math. Soc. {42} (1974), 575-580)</a>, Proc. Amer. Math. Soc. 48 (1975), 518.
%H Khristo N. Boyadzhiev, <a href="https://arxiv.org/abs/1806.09468">Close encounters with the Stirling numbers of the second kind</a>, arXiv:1806.09468 [math.HO], 2018.
%H B. A. Bondarenko, <a href="http://www.fq.math.ca/pascal.html">Generalized Pascal Triangles and Pyramids</a>, English translation published by Fibonacci Association, Santa Clara Univ., Santa Clara, CA, 1993; see p. 42.
%H S. Alex Bradt, Jennifer Elder, Pamela E. Harris, Gordon Rojas Kirby, Eva Reutercrona, Yuxuan (Susan) Wang, and Juliet Whidden, <a href="https://arxiv.org/abs/2401.06937">Unit interval parking functions and the r-Fubini numbers</a>, arXiv:2401.06937 [math.CO], 2024. See page 8.
%H Pascal Caron, Jean-Gabriel Luque, Ludovic Mignot, and Bruno Patrou, <a href="http://arxiv.org/abs/1505.03474">State complexity of catenation combined with a boolean operation: a unified approach</a>, arXiv:1505.03474 [cs.FL], 2015.
%H J. L. Cereceda, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Cereceda/cereceda2.html">Generalized Akiyama-Tanigawa Algorithm for Hypersums of Powers of Integers</a>, J. Int. Seq. 16 (2013) #13.3.2.
%H Raphaël Cerf and Joseba Dalmau, <a href="https://arxiv.org/abs/1609.05738">The quasispecies distribution</a>, arXiv:1609.05738 [q-bio.PE], 2016.
%H Gi-Sang Cheon and Jin-Soo Kim, <a href="https://doi.org/10.1016/S0024-3795(01)00234-8">Stirling matrix via Pascal matrix</a>, Lin. Alg. Appl. 329 (1-3) (2001) 49-59
%H Sarthak Chimni and Ramin Takloo-Bighash, <a href="https://arxiv.org/abs/1812.09564">Counting subrings of Zn of non-zero co-rank</a>, arXiv:1812.09564 [math.NT], 2018.
%H C. Cooper and R. E. Kennedy, <a href="http://cs.ucmo.edu/~cnc8851/articles/pasnsk.pdf">Patterns, automata and Stirling numbers of the second kind</a>, Mathematics and Computer Education Journal, 26 (1992), 120-124.
%H T. Copeland, <a href="https://tcjpn.wordpress.com/2020/08/11/euler-bernoulli-worpitzky-identities/">Reciprocity and Umbral Witchcraft: An Eve with Stirling, Bernoulli, Archimedes, Euler, Laguerre, and Worpitzky</a>, 2020.
%H T. Copeland's Shadows of Simplicity, <a href="http://tcjpn.wordpress.com/2015/08/23/a-class-of-differential-operators-and-the-stirling-numbers/">A Class of Differential Operators and the Stirling Numbers</a>,2015; <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, 2015; <a href="http://tcjpn.wordpress.com/2011/11/16/a-generalized-dobinski-relation-and-the-confluent-hypergeometric-fcts/">The Inverse Mellin Transform, Bell Polynomials, a Generalized Dobinski Relation, and the Confluent Hypergeometric Functions</a>, 2011; <a href="http://tcjpn.wordpress.com/2008/06/12/mathemagical-forests/">Mathemagical Forests</a>, 2008; and <a href="http://tcjpn.wordpress.com/2010/12/28/14/">Addendum to "Mathemagical Forests"</a>, 2010.
%H R. M. Dickau, <a href="http://mathforum.org/advanced/robertd/stirling2.html">Stirling numbers of the second kind</a>
%H A. J. Dobson, <a href="https://doi.org/10.1016/S0021-9800(68)80060-2">A note on Stirling numbers of the second kind</a>, Journal of Combinatorial Theory 5.2 (1968): 212-214.
%H Tomislav Došlic and Darko Veljan, <a href="http://dx.doi.org/10.1016/j.disc.2007.04.066">Logarithmic behavior of some combinatorial sequences</a>, Discrete Math. 308 (2008), no. 11, 2182--2212. MR2404544 (2009j:05019)
%H G. Duchamp, K. A. Penson, A. I. Solomon, A. Horzela and P. Blasiak, <a href="http://arXiv.org/abs/quant-ph/0401126">One-parameter groups and combinatorial physics</a>, arXiv:quant-ph/0401126, 2004.
%H Askar Dzhumadil’daev and Damir Yeliussizov, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v22i4p10">Walks, partitions, and normal ordering</a>, Electronic Journal of Combinatorics, 22(4) (2015), #P4.10.
%H FindStat - Combinatorial Statistic Finder, <a href="http://www.findstat.org/St000105/">The number of blocks in the set partition</a>
%H Ghislain R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H M. L. Glasser, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Glasser/glasser2.html">A Generalized Apery Series</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.4.3.
%H Bill Gosper, <a href="/A008277/a008277.png">Colored illustrations of triangle of Stirling numbers of second kind read mod 2, 3, 4, 5, 6, 7</a>
%H M. Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Griffiths2/griffiths17.html">Remodified Bessel Functions via Coincidences and Near Coincidences</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.7.1.
%H M. Griffiths, <a href="http://www.jstor.org/stable/10.5951/mathteacher.106.4.0313">Close Encounters with Stirling Numbers of the Second Kind</a>, The Mathematics Teacher, Vol. 106, No. 4, November 2012, pp. 313-317.
%H M. Griffiths and I. Mezo, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Griffiths/griffiths11.html">A generalization of Stirling Numbers of the Second Kind via a special multiset</a>, JIS 13 (2010) #10.2.5
%H J. Gubeladze and J. Love, <a href="http://arxiv.org/abs/1304.3775">Vertex maps between simplices, cubes, and crosspolytopes</a>, arXiv:1304.3775 [math.CO], 2013.
%H L. C. Hsu, <a href="http://dx.doi.org/10.1214/aoms/1177730254">Note on an asymptotic expansion of the n-th difference of zero</a>, Ann. Math. Statistics 19 (1948), 273-277.
%H Yoshinari Inaba, <a href="http://www.emis.de/journals/JIS/VOL8/Inaba/inaba301.html">Hyper-Sums of Powers of Integers and the Akiyama-Tanigawa Matrix</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.7.
%H Wayne A. Johnson, <a href="https://arxiv.org/abs/1804.04943">Exponential Hilbert series of equivariant embeddings</a>, arXiv:1804.04943 [math.RT], 2018.
%H Matthieu Josuat-Verges, <a href="http://igm.univ-mlv.fr/~josuatv/files/josuat-qstirling.pdf">A q-analog of Schläfli and Gould identities on Stirling numbers</a>, Preprint, 2016; also arXiv:1610.02965 [math.CO], 2016.
%H Charles Knessl and Joseph B. Keller, <a href="http://dx.doi.org/10.1002/sapm199184143">Stirling number asymptotics from recursion equations using the ray method</a>, Stud. Appl. Math. 84 (1991), no. 1, 43-56.
%H Nate Kube and Frank Ruskey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL8/Ruskey/ruskey99.html">Sequences That Satisfy a(n-a(n))=0</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.5.5.
%H D. E. Knuth, <a href="http://arxiv.org/abs/math/9207221">Convolution polynomials</a>, The Mathematica J., 2 (1992), 67-78.
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H Elliott H. Lieb, <a href="https://doi.org/10.1016/S0021-9800(68)80057-2">Concavity properties and a generating function for Stirling numbers</a>, Journal of Combinatorial Theory, Vol. 5, No. 2 (1968), pp. 203-206.
%H Shi-Mei Ma, <a href="http://arxiv.org/abs/1208.3104">Some combinatorial sequences associated with context-free grammars</a>, arXiv:1208.3104v2 [math.CO], 2012.
%H Shi-Mei Ma, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v20i1p11">A family of two-variable derivative polynomials for tangent and secant</a>, El J. Combinat. 20 (1) (2013) P11
%H S.-M. Ma, Toufik Mansour, and Matthias Schork. <a href="http://arxiv.org/abs/1308.0169">Normal ordering problem and the extensions of the Stirling grammar</a>, arXiv:1308.0169 [math.CO], 2013.
%H M. M. Mangontarum and J. Katriel, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Mangontarum/mango2.html">On q-Boson Operators and q-Analogues of the r-Whitney and r-Dowling Numbers</a>, J. Int. Seq. 18 (2015) 15.9.8.
%H T. Manneville and V. Pilaud, <a href="http://arxiv.org/abs/1501.07152">Compatibility fans for graphical nested complexes</a>, arXiv:1501.07152 [math.CO], 2015.
%H Toufik Mansour, A. Munagi, and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Mansour/mansour4.html">Recurrence Relations and Two-Dimensional Set Partitions </a>, J. Int. Seq. 14 (2011) # 11.4.1
%H Toufik Mansour and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Shattuck/shattuck3.html">Counting Peaks and Valleys in a Partition of a Set</a>, J. Int. Seq. 13 (2010), 10.6.8.
%H Toufik Mansour, Matthias Schork and Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Schork/schork2.html">The Generalized Stirling and Bell Numbers Revisited</a>, Journal of Integer Sequences, Vol. 15 (2012), #12.8.3.
%H Richard J. Mathar, <a href="https://arxiv.org/abs/1903.12477">2-regular Digraphs of the Lovelock Lagrangian</a>, arXiv:1903.12477 [math.GM], 2019.
%H Nelma Moreira and Rogerio Reis, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Moreira/moreira8.html">On the Density of Languages Representing Finite Set Partitions</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.2.8.
%H T. S. Motzkin, <a href="/A000262/a000262.pdf">Sorting numbers for cylinders and other classification numbers</a>, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
%H A. O. Munagi, <a href="http://dx.doi.org/10.1155/IJMMS.2005.215">k-Complementing Subsets of Nonnegative Integers</a>, International Journal of Mathematics and Mathematical Sciences, 2005:2 (2005), 215-224.
%H Emanuele Munarini, <a href="https://doi.org/10.2298/AADM180226017M">Combinatorial identities involving the central coefficients of a Sheffer matrix</a>, Applicable Analysis and Discrete Mathematics (2019) Vol. 13, 495-517.
%H Norihiro Nakashima and Shuhei Tsujie, <a href="https://arxiv.org/abs/1904.09748">Enumeration of Flats of the Extended Catalan and Shi Arrangements with Species</a>, arXiv:1904.09748 [math.CO], 2019.
%H G. Nemes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Nemes/nemes2.html">On the Coefficients of the Asymptotic Expansion of n!</a>, J. Int. Seq. 13 (2010), 10.6.6.
%H A. F. Neto, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Neto/neto4.html">Higher Order Derivatives of Trigonometric Functions, Stirling Numbers of the Second Kind, and Zeon Algebra</a>, Journal of Integer Sequences, Vol. 17 (2014), Article 14.9.3.
%H Arthur Nunge, <a href="https://arxiv.org/abs/1805.01797">Eulerian polynomials on segmented permutations</a>, arXiv:1805.01797 [math.CO], 2018.
%H OEIS Wiki, <a href="http://oeis.org/wiki/Sorting_numbers">Sorting numbers</a>
%H K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, <a href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type relations via substitution and the moment problem</a>, arXiv:quant-ph/0312202, 2003.
%H K. A. Penson, P. Blasiak, A. Horzela, G. H. E. Duchamp, and A. I. Solomon, <a href="http://arxiv.org/abs/0904.0369">Laguerre-type derivatives: Dobinski relations and combinatorial identities</a>, J. Math. Phys. vol. 50, 083512 (2009)
%H Mathias Pétréolle and Alan D. Sokal, <a href="https://arxiv.org/abs/1907.02645">Lattice paths and branched continued fractions. II. Multivariate Lah polynomials and Lah symmetric functions</a>, arXiv:1907.02645 [math.CO], 2019.
%H C. J. Pita Ruiz V., <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Pita/pita19.html">Some Number Arrays Related to Pascal and Lucas Triangles</a>, J. Int. Seq. 16 (2013) #13.5.7
%H Feng Qi, <a href="http://arxiv.org/abs/1402.2361">An Explicit Formula for Bell Numbers in Terms of Stirling Numbers and Hypergeometric Functions</a>, arXiv:1402.2361 [math.CO], 2014.
%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/NoteBooks/NoteBook2/chapterIII/page4.htm">Notebook entry</a>
%H René Rietz, <a href="https://opus4.kobv.de">Optimization of Network Intrusion Detection Processes</a>, 2018.
%H G. Rzadkowski, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Rzadkowski/rzadkowski3.html">Two formulas for Successive Derivatives and Their Applications</a>, JIS 12 (2009) 09.8.2
%H Benjamin Schreyer, <a href="https://arxiv.org/abs/2409.03799">Rigged Horse Numbers and their Modular Periodicity</a>, arXiv:2409.03799 [math.CO], 2024. See p. 12.
%H Raymond Scurr and Gloria Olive, <a href="http://dx.doi.org/10.1016/S0012-365X(98)00011-9">Stirling numbers revisited</a>, Discrete Math. 189 (1998), no. 1-3, 209--219. MR1637761 (99d:11019).
%H Mark Shattuck, <a href="https://www.researchgate.net/publication/269168303_Combinatorial_proofs_of_some_Stirling_number_formulas">Combinatorial proofs of some Stirling number formulas</a>, Preprint (ResearchGate), 2014.
%H Mark Shattuck, <a href="http://dx.doi.org/10.1515/puma-2015-0009">Combinatorial proofs of some Stirling number formulas</a>, Pure Mathematics and Applications, Volume 25, Issue 1 (Sep 2015).
%H Mark Shattuck, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Shattuck/sha27.html">Combinatorial Proofs of Some Stirling Number Convolution Formulas</a>, J. Int. Seq., Vol. 25 (2022), Article 22.2.2.
%H John K. Sikora, <a href="https://arxiv.org/abs/1806.00887">On Calculating the Coefficients of a Polynomial Generated Sequence Using the Worpitzky Number Triangles</a>, arXiv:1806.00887 [math.NT], 2018.
%H A. I. Solomon, P. Blasiak, G. Duchamp, A. Horzela, and K. A. Penson, <a href="http://arXiv.org/abs/quant-ph/0409082">Partition functions and graphs: A combinatorial approach</a>, arXiv:quant-ph/0409082, 2004.
%H M. Z. Spivey, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Spivey/spivey31.html">On Solutions to a General Combinatorial Recurrence</a>, J. Int. Seq. 14 (2011) # 11.9.7.
%H Jacob Sprittulla, <a href="https://arxiv.org/abs/2109.12705">The ordered Bell numbers as weighted sums of odd or even Stirling numbers of the second kind</a>, arXiv:2109.12705 [math.CO], 2021.
%H N. M. Temme, <a href="http://oai.cwi.nl/oai/asset/2304/2304A.pdf">Asymptotic estimates of Stirling numbers</a>, Stud. Appl. Math. 89 (1993), no. 3, 233-243.
%H A. N. Timashev, <a href="http://dx.doi.org/10.4213/dm440">On asymptotic expansions of Stirling numbers of the first and second kinds</a>. (Russian) Diskret. Mat. 10 (1998), no. 3,148-159 translation in Discrete Math. Appl. 8 (1998), no. 5, 533-544.
%H Michael Torpey, <a href="https://doi.org/10.17630/10023-17350">Semigroup congruences: computational techniques and theoretical applications</a>, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).
%H A. H. Voigt, <a href="/A000918/a000918.pdf">Theorie der Zahlenreihen und der Reihengleichungen</a>, Goschen, Leipzig, 1911. [Annotated scans of pages 30-33 only]
%H Dennis Walsh, <a href="http://frank.mtsu.edu/~dwalsh/STIRFORT.pdf">Counting forests with Stirling and Bell numbers</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DifferentialOperator.html">Differential Operator</a> and <a href="http://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Stirling Number of the Second Kind</a>
%H Thomas Wieder, The number of certain k-combinations of an n-set, <a href="http://www.math.nthu.edu.tw/~amen/">Applied Mathematics Electronic Notes</a>, vol. 8 (2008).
%H H. S. Wilf, <a href="http://www.math.upenn.edu/~wilf/DownldGF.html">Generatingfunctionology</a>, 2nd edn., Academic Press, NY, 1994, pp. 17ff, 105ff.
%H M. C. Wolf, <a href="http://dx.doi.org/10.1215/S0012-7094-36-00253-3">Symmetric functions for non-commutative elements</a>, Duke Math. J., 2 (1936), 626-637.
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F 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.
%F E.g.f.: A(x, y) = e^(y*e^x-y). E.g.f. for m-th column: (e^x-1)^m/m!.
%F S2(n, k) = (1/k!) * Sum_{i=0..k} (-1)^(k-i)*binomial(k, i)*i^n.
%F Row sums: Bell number A000110(n) = Sum_{k=1..n} S2(n, k), n>0.
%F S(n, k) = Sum (i_1*i_2*...*i_(n-k)) summed over all (n-k)-combinations {i_1, i_2, ..., i_k} with repetitions of the numbers {1, 2, ..., k}. Also S(n, k) = Sum (1^(r_1)*2^(r_2)*...* k^(r_k)) summed over integers r_j >= 0, for j=1..k, with Sum{j=1..k} r_j = n-k. [Charalambides]. - _Wolfdieter Lang_, Aug 15 2019.
%F A019538(n, k) = k! * S2(n, k).
%F A028248(n, k) = (k-1)! * S2(n, k).
%F For asymptotics see Hsu (1948), among other sources.
%F Sum_{n>=0} S2(n, k)*x^n = x^k/((1-x)(1-2x)(1-3x)...(1-kx)).
%F Let P(n) = the number of integer partitions of n (A000041), p(i) = the number of parts of the i-th partition of n, d(i) = the number of distinct 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, and Sum_{i=1..P(n), p(i)=m} = sum running from i=1 to i=P(n) but taking only partitions with p(i)=m parts into account. Then S2(n, m) = Sum_{i=1..P(n), p(i)=m} n!/(Product_{j=1..p(i)} p(i, j)!) * 1/(Product_{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_, Jun 02 2005
%F Sum_{k=1..n} k*S2(n,k) = B(n+1)-B(n), where B(q) are the Bell numbers (A000110). - _Emeric Deutsch_, Nov 01 2006
%F Recurrence: S2(n+1,k) = Sum_{i=0..n} binomial(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} binomial(n-1,i-1)*S2(i-1,k-1). - _Thomas Wieder_, Jan 27 2007
%F 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): S2(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: S2(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_, Mar 28 2007
%F From _Tom Copeland_, Oct 10 2007: (Start)
%F 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) * binomial(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; and Lag(n,x,m), the associated Laguerre polynomials of order m.
%F For x = 0, the equation gives Sum_{j=0..n} E(n,j) * binomial(j,n) = 1 for n=0 and 0 for all other n. By substituting the umbral compositional inverse of the Bell polynomials, the lower factorial n!*binomial(y,n), for x in the equation, the Worpitzky identity is obtained; y^n = Sum_{j=0..n} E(n,j) * binomial(y+j,n).
%F 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)).
%F (Appended Sep 16 2020) For connections to the Bernoulli numbers, extensions, proofs, and a clear presentation of the number arrays involved in the identities above, see my post Reciprocity and Umbral Witchcraft. (End)
%F 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_, Nov 21 2007
%F The row polynomials are given by D^n(e^(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
%F Let f(x) = e^(e^x). Then for n >= 1, 1/f(x)*(d/dx)^n(f(x)) = 1/f(x)*(d/dx)^(n-1)(e^x*f(x)) = Sum_{k=1..n} S2(n,k)*e^(k*x). Similar formulas hold for A039755, A105794, A111577, A143494 and A154537. - _Peter Bala_, Mar 01 2012
%F S2(n,k) = A048993(n,k), 1 <= k <= n. - _Reinhard Zumkeller_, Mar 26 2012
%F O.g.f. for the n-th diagonal is D^n(x), where D is the operator x/(1-x)*d/dx. - _Peter Bala_, Jul 02 2012
%F n*i!*S2(n-1,i) = Sum_{j=(i+1)..n} (-1)^(j-i+1)*j!/(j-i)*S2(n,j). - _Leonid Bedratyuk_, Aug 19 2012
%F G.f.: (1/Q(0)-1)/(x*y), where Q(k) = 1 - (y+k)*x - (k+1)*y*x^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Nov 09 2013
%F From _Tom Copeland_, Apr 17 2014: (Start)
%F Multiply each n-th diagonal of the Pascal lower triangular matrix by x^n and designate the result as A007318(x) = P(x).
%F With Bell(n,x)=B(n,x) defined above, D = d/dx, and :xD:^n = x^n*D^n, a Dobinski formula gives umbrally f(y)^B(.,x) = e^(-x)*e^(f(y)*x). Then f(y)^B(.,:xD:)g(x) = [f(y)^(xD)]g(x) = e^[-(1-f(y)):xD:]g(x) = g[f(y)x].
%F In particular, for f(y) = (1+y),
%F A) (1+y)^B(.,x) = e^(-x)*e^((1+y)*x) = e^(x*y) = e^[log(1+y)B(.,x)],
%F B) (I+dP)^B(.,x) = e^(x*dP) = P(x) = e^[x*(e^M-I)]= e^[M*B(.,x)] with dP = A132440, M = A238385-I = log(I+dP), and I = identity matrix, and
%F C) (1+dP)^(xD) = e^(dP:xD:) = P(:xD:) = e^[(e^M-I):xD:] = e^[M*xD] with action e^(dP:xD:)g(x) = g[(I+dP)*x].
%F D) P(x)^m = P(m*x), which implies (Sum_{k=1..m} a_k)^j = B(j,m*x) where the sum is umbrally evaluated only after exponentiation with (a_k)^q = B(.,x)^q = B(q,x). E.g., (a1+a2+a3)^2=a1^2+a2^2+a3^2+2(a1*a2+a1*a3+a2*a3) = 3*B(2,x)+6*B(1,x)^2 = 9x^2+3x = B(2,3x).
%F E) P(x)^2 = P(2x) = e^[M*B(.,2x)] = A038207(x), the face vectors of the n-Dim hypercubes.
%F (End)
%F As a matrix equivalent of some inversions mentioned above, A008277*A008275 = I, the identity matrix, regarded as lower triangular matrices. - _Tom Copeland_, Apr 26 2014
%F O.g.f. for the n-th diagonal of the triangle (n = 0,1,2,...): Sum_{k>=0} k^(k+n)*(x*e^(-x))^k/k!. Cf. the generating functions of the diagonals of A039755. Also cf. A112492. - _Peter Bala_, Jun 22 2014
%F Floor(1/(-1 + Sum_{n>=k} 1/S2(n,k))) = A034856(k-1), for k>=2. The fractional portion goes to zero at large k. - _Richard R. Forberg_, Jan 17 2015
%F From _Daniel Forgues_, Jan 16 2016: (Start)
%F Let x_(n), called a factorial term (Boole, 1970) or a factorial polynomial (Elaydi, 2005: p. 60), denote the falling factorial Product_{k=0..n-1} (x-k). Then, for n >= 1, x_(n) = Sum_{k=1..n} A008275(n,k) * x^k, x^n = Sum_{k=1..n} T(n,k) * x_(k), where A008275(n,k) are Stirling numbers of the first kind.
%F For n >= 1, the row sums yield the exponential numbers (or Bell numbers): Sum_{k=1..n} T(n,k) = A000110(n), and Sum_{k=1..n} (-1)^(n+k) * T(n,k) = (-1)^n * Sum_{k=1..n} (-1)^k * T(n,k) = (-1)^n * A000587(n), where A000587 are the complementary Bell numbers. (End)
%F Sum_{k=1..n} k*S2(n,k) = A138378(n). - _Alois P. Heinz_, Jan 07 2022
%F O.g.f. for the m-th column: x^m/(Product_{j=1..m} 1-j*x). - _Daniel Checa_, Aug 25 2022
%F S2(n,k) ~ (k^n)/k!, for fixed k as n->oo. - _Daniel Checa_, Nov 08 2022
%e The triangle S2(n, k) begins:
%e \ k 1 2 3 4 5 6 7 8 9
%e n \ 10 11 12 13 14 15 ...
%e ----------------------------------------------------------------------------------
%e 1 | 1
%e 2 | 1 1
%e 3 | 1 3 1
%e 4 | 1 7 6 1
%e 5 | 1 15 25 10 1
%e 6 | 1 31 90 65 15 1
%e 7 | 1 63 301 350 140 21 1
%e 8 | 1 127 966 1701 1050 266 28 1
%e 9 | 1 255 3025 7770 6951 2646 462 36 1
%e 10 | 1 511 9330 34105 42525 22827 5880 750 45
%e 1
%e 11 | 1 1023 28501 145750 246730 179487 63987 11880 1155
%e 55 1
%e 12 | 1 2047 86526 611501 1379400 1323652 627396 159027 22275
%e 1705 66 1
%e 13 | 1 4095 261625 2532530 7508501 9321312 5715424 1899612 359502
%e 39325 2431 78 1
%e 14 | 1 8191 788970 10391745 40075035 63436373 49329280 20912320 5135130
%e 752752 66066 3367 91 1
%e 15 | 1 16383 2375101 42355950 210766920 420693273 408741333 216627840 67128490
%e 12662650 1479478 106470 4550 105 1
%e ...
%e ----------------------------------------------------------------------------------
%e x^4 = 1 x_(1) + 7 x_(2) + 6 x_(3) + 1 x_(4), where x_(k) = P(x,k) = k!*C(x,k). - _Daniel Forgues_, Jan 16 2016
%p seq(seq(combinat[stirling2](n, k), k=1..n), n=1..10); # _Zerinvary Lajos_, Jun 02 2007
%p stirling_2 := (n,k) -> (1/k!) * add((-1)^(k-i)*binomial(k,i)*i^n, i=0..k);
%t Table[StirlingS2[n, k], {n, 11}, {k, n}] // Flatten (* _Robert G. Wilson v_, May 23 2006 *)
%t BellMatrix[f_, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t rows = 12;
%t B = BellMatrix[1&, rows];
%t Table[B[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 28 2018, after _Peter Luschny_ *)
%t a[n_, n_] := 1; a[n_, 1] := 1;
%t a[n_, k_] := a[n, k] = a[n-1, k-1] + k a[n-1, k]; Flatten@
%t Table[a[n, k], {n, 1, 11}, {k, 1, n}] (* _Oliver Seipel_, Jun 12 2024 *)
%t With[{m = 11},
%t Flatten@MapIndexed[Take[#, #2[[1]]] &,
%t Transpose@
%t Table[Range[1, m]! Coefficient[(E^x-1)^k/k! + O[x]^(m+1), x,
%t Range[1, m]], {k, 1, m}]]] (* _Oliver Seipel_, Jun 12 2024 *)
%o (PARI) for(n=1,22,for(k=1,n,print1(stirling(n,k,2),", "));print()); \\ _Joerg Arndt_, Apr 21 2013
%o (PARI) Stirling2(n,k)=sum(i=0,k,(-1)^i*binomial(k,i)*i^n)*(-1)^k/k! \\ _M. F. Hasler_, Mar 06 2012
%o (Haskell)
%o a008277 n k = a008277_tabl !! (n-1) !! (k-1)
%o a008277_row n = a008277_tabl !! (n-1)
%o a008277_tabl = map tail $ a048993_tabl -- _Reinhard Zumkeller_, Mar 26 2012
%o (Maxima) create_list(stirling2(n+1,k+1),n,0,30,k,0,n); /* _Emanuele Munarini_, Jun 01 2012 */
%o (Sage) stirling_number2 # _Danny Rorabaugh_, Oct 11 2015
%o (J) n ((] (1 % !)) * +/@((^~ * (] (_1 ^ |.)) * (! {:)@]) i.@>:)) k NB. _Stephen Makdisi_, Apr 06 2016
%o (Magma) [[StirlingSecond(n,k): k in [1..n]]: n in [1..12]]; // _G. C. Greubel_, May 22 2019
%Y Cf. A008275 (Stirling numbers of first kind), A048993 (another version of this triangle).
%Y Cf. A000217, A001296, A001297, A001298, A007318, A028246, A039810-A039813, A048994, A087107-A087111, A087127, A094262, A127701.
%Y See also A331155.
%Y Cf. A038207, A086885, A132440, A138378, A173018, A238385.
%Y Cf. A102661 (partial row sums).
%K nonn,easy,tabl,nice,core,changed
%O 1,5
%A _N. J. A. Sloane_