

A019538


Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n).


105



1, 1, 2, 1, 6, 6, 1, 14, 36, 24, 1, 30, 150, 240, 120, 1, 62, 540, 1560, 1800, 720, 1, 126, 1806, 8400, 16800, 15120, 5040, 1, 254, 5796, 40824, 126000, 191520, 141120, 40320, 1, 510, 18150, 186480, 834120, 1905120, 2328480, 1451520, 362880, 1, 1022, 55980
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OFFSET

1,3


COMMENTS

Number of ways n labeled objects can be distributed into k nonempty parcels. Also number of special terms in n variables with maximal degree k.
In older terminology these are called differences of 0.  Michael Somos, Oct 08 2003
Number of surjections (onto functions) from an nelement set to a kelement set.
Also coefficients (in ascending order) of socalled ordered Bell polynomials.
(k1)!*Stirling2(n,k1) is the number of chain topologies on an nset having k open sets [Stephen].
Number of set compositions (ordered set partitions) of n items into k parts. Number of k dimensional 'faces' of the n dimensional permutohedron (see Simion, p. 162).  Mitch Harris, Jan 16 2007
Correction of comment before: Number of (nk)dimensional 'faces' of the permutohedron of order n (an (n1)dimensional polytope).  Tilman Piesk, Oct 29 2014
This array is related to the reciprocal of an e.g.f. as sketched in A133314. For example, the coefficient of the fourth order term in the Taylor series expansion of 1/(a(0) + a(1) x + a(2) x^2/2! + a(3) x^3/3! + ...) is a(0)^(5) * {24 a(1)^4  36 a(1)^2 a(2) a(0) + [8 a(1) a(3) + 6 a(2)^2] a(0)^2  a(4) a(0)^3}. The unsigned coefficients characterize the P3 permutohedron depicted on page 10 in the Loday link with 24 vertices (0D faces), 36 edges (1D faces), 6 squares (2D faces), 8 hexagons (2D faces) and 1 3D permutohedron. Summing coefficients over like dimensions gives A019538 and A090582. Compare to A133437 for the associahedron.  Tom Copeland, Sep 29 2008, Oct 07 2008
Further to the comments of Tom Copeland above, the permutohedron of type A_3 can be taken as the truncated octahedron. Its dual is the tetrakis hexahedron, a simplicial polyhedron, with fvector (1,14,36,24) giving the fourth row of this triangle. See the Wikipedia entry and [Fomin and Reading p. 21]. The corresponding hvectors of permutohedra of type A give the rows of the triangle of Eulerian numbers A008292. See A145901 and A145902 for the array of fvectors for type B and type D permutohedra respectively.  Peter Bala, Oct 26 2008
Subtriangle of triangle in A131689.  Philippe Deléham, Nov 03 2008
Since T(n,k) counts surjective functions and surjective functions are "consistent", T(n,k) satisfies a binomial identity, namely, T(n,x+y)=sum(C(n,j)*T(j,x)*T(nj,y), j=0..n). For definition of consistent functions and a generalized binomial identity, see "Toy stories and combinatorial identities" in the link section below.  Dennis P. Walsh, Feb 24 2012
T(n,k) is the number of labeled forests on n+k vertices satisfying the following two conditions: (i) each forest consists of exactly k rooted trees with roots labeled 1, 2,...,k; (ii) every root has at least one child vertex.  Dennis P. Walsh, Feb 24 2012
The triangle is the inverse binomial transform of triangle A028246, deleting the left column and shifting up one row.  Gary W. Adamson, Mar 05 2012
See A074909 for associations among this array and the Bernoulli polynomials and their umbral compositional inverses.  Tom Copeland, Nov 14 2014
E.g.f. for the shifted signed polynomials is G(x,t) = (e^t1)/[1+(1+x)(e^t1)] = 1(1+x)(e^t1) + (1+x)^2(e^t1)^2  ... (see also A008292 and A074909), which has the infinitesimal generator g(x,u)d/du = [(1x*u)(1(1+x)u)]d/du, i.e., exp[t*g(x,u)d/du]u eval. at u=0 gives G(x,t), and dG(x,t)/dt=g(x,G(x,t)). The compositional inverse is log((1xt)/(1(1+x)t)). G(x,t) is a generating series associated to the generalized Hirzebruch genera. See the G. Rzadowski link for the relation of the derivatives of g(x,u) to solutions of the Riccatt differential equation, soliton solns. to the KdV equation, and the Eulerian and Bernoulli numbers. In addition A145271 connects products of derivatives of g(x,u) and the refined Eulerian numbers to the inverse of G(x,t), which gives the normalized, reverse face polynomials of the simplices (A135278, divided by n+1). See A028246 for the generator g(x,u)d/dx.  Tom Copeland, Nov 21 2014
For connections to toric varieties and Eulerian polynomials, see the Dolgachev and Lunts and the Stembridge links.  Tom Copeland, Dec 31 2015


REFERENCES

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, p. 89, ex. 1; also p. 210.
Moussa Benoumhani, The number of topologies on a finite set, Preprint, 2005.
Miklos Bona, Combinatorics of Permutations, Chapman and Hall,2004, p.12.
G. Boole, A Treatise On The Calculus of Finite Differences, Dover Publications, 1960, p. 20.
H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 212.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. AddisonWesley, Reading, 1989, p. 155.
Kiran S. Kedlaya and Andrew V. Sutherland Computing L Series of Hyperelliptic Curves in Algorithmic Number Theory Lecture Notes in Computer Science Volume 5011/2008
T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 5.6.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.
Mark Shattuck, Combinatorial proofs of some Stirling number formulas, Preprint (ResearchGate), 2014.
J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31.
E. Whittaker and G. Robinson, The Calculus of Observations, Blackie, London, 4th ed., 1949; p. 7.


LINKS

T. D. Noe, Rows n=1..100 of triangle, flattened
J. Fernando Barbero G., Jesús Salas, Eduardo J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. I. General Structure, arXiv:1307.2010 [math.CO], 2013.
J. F. Barbero G., J. Salas and E. J. S. Villaseñor, Bivariate Generating Functions for a Class of Linear Recurrences. II. Applications, arXiv preprint arXiv:1307.5624 [math.CO], 2013.
K. N. Boyadzhiev, A Series transformation formula and related polynomials, Int. J. Math. Math. Sc. 23 (2005), 38493866.
J. L. Chandon, J. LeMaire and J. Pouget, Dénombrement des quasiordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 6180.
T. Copeland, The Bernoulli polynomials and Hirzebruch's generalized Todd class
E. Delucchi, A. Pixton and L. Sabalka, Face vectors of subdivided simplicial complexes Discrete Math. 312 (2012), no. 2, 248257. MR2852583 (2012j:05470). See p. 250.  N. J. A. Sloane, Apr 04 2014
I. Dolgachev and V. Lunts, A character formula for the representation of the Weyl group in the cohomology of the associated toric variety, Journal of Algebra, 168, 741772, (1994).
S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/ParkCity 2004. [From Peter Bala, Oct 26 2008]
M. Goebel, On the number of special permutationinvariant orbits and terms, in Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505509 (Lect. Notes Comp. Sci.)
M. Griffiths, I. Mezo, A generalization of Stirling Numbers of the Second Kind via a special multiset, JIS 13 (2010) #10.2.5
J. Gubeladze and J. Love, Vertex maps between simplices, cubes, and crosspolytopes, arXiv preprint arXiv:1304.3775 [math.CO], 2013.
Li Guo, Baxter algebras, Stirling numbers and partitions, arXiv:math/0402348 [math.AC], 2004.
M. Iida, On Triangle of numbers, Josai Mathematical Monographs, Vol. 5 (2012), 6170;
Gábor Hetyei, The Stirling polynomial of a simplicial complex Discrete and Computational Geometry 35, Number 3, March 2006, pp 437455. [From Peter Bala, Oct 28 2008]
Gábor Hetyei, Face enumeration using generalized binomial coefficients. Online draft version of the Hetyei paper referenced above. [From Peter Bala, Nov 10 2008]
Germain Kreweras, Une dualité élémentaire souvent utile dans les problèmes combinatoires, Mathématiques et Sciences Humaines 3 (1963): 3141.
G. Kreweras, Les preordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 530.
L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 20052006. [From Peter Bala, Oct 26 2008]
J. Loday, The Multiple Facets of the Associahedron [From Tom Copeland, Sep 29 2008]
MathOverflow, How many surjections are there from a set of size n? [From Tom Copeland, Sep 09 2015]
E. Mendelson, Races with Ties, Math. Mag. 55 (1982), 170175.
T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167176. [Annotated, scanned copy]
OEIS Wiki, Sorting numbers
K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinskitype relations via substitution and the moment problem, arXiv:quantph/0312202, 2003.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257260.
P. A. Piza, Kummer numbers, Mathematics Magazine, 21 (1947/1948), 257260. [Annotated scanned copy]
G. Rzadkowski, Bernoulli numbers and solitons revisited, Journal of Nonlinear Mathematical Physics, 17:1, 121126, DOI: 10.1142/S1402925110000635
R. Simion, Convex Polytopes and Enumeration, Adv. in Appl. Math. 18 (1997) pp. 149180.
J. Stembridge, Some permuutation representations of Weyl groups associated with the cohomology of toric varieties, Adv. in Math., 106, 244301, (1994).
D. Stephen, Topology on Finite Sets, American Mathematical Monthly, 75: 739  741, 1968.
S. M. Tanny, On some numbers related to the Bell numbers, Canad. Math. Bull. Vol. 17 (5), 733738, 1975
A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911. [Annotated scans of pages 3033 only]
D. P. Walsh, Toy stories and combinatorial identities
Wikipedia, Truncated octahedron [From Peter Bala, Oct 26 2008]


FORMULA

T(n, k) = k*(T(n1, k1)+T(n1, k)) with T(0, 0) = 1 [or T(1, 1) = 1].  Henry Bottomley, Mar 02 2001
E.g.f.: (y*(exp(x)1)  exp(x))/(y*(exp(x)1)  1).  Vladeta Jovovic, Jan 30 2003
Equals [0, 1, 0, 2, 0, 3, 0, 4, 0, 5, ...] DELTA [1, 1, 2, 2, 3, 3, 4, 4, 5, 5, ...] where DELTA is Deléham's operator defined in A084938.
T(n, k) = Sum_{j=0..k} (1)^(kj)*j^n*binomial(k, j).  Mario Catalani (mario.catalani(AT)unito.it), Nov 28 2003. See Graham et al., eq. (6.19), p. 251. For a proof see Bert Seghers, Jun 29 2013.
Sum_{k=0..n} T(n, k)(1)^(nk) = 1, Sum_{k=0..n} T(n, k)(1)^k = (1)^n.  Mario Catalani (mario.catalani(AT)unito.it), Dec 11 2003
O.g.f. for nth row: polylog(n, x/(1+x))/(x+x^2).  Vladeta Jovovic, Jan 30 2005
E.g.f.: 1 / {1 + t[1exp(x)]}.  Tom Copeland, Oct 13 2008
From Peter Bala, Oct 26 2008: (Start)
O.g.f. as a continued fraction: 1/(1  x*t/(1  (x + 1)*t/(1  2*x*t/(1  2*(x + 1)*t/(1  ...))))) = 1 + x*t + (x + 2*x^2)*t^2 + (x + 6*x^2 + 6*x^3)*t^3 + ... .
The row polynomials R(n,x), which begin R(1,x) = x, R(2,x) = x + 2*x^2, R(3,x) = x + 6*x^2 + 6*x^3, satisfy the recurrence x*d/dx ((x + 1)*R(n,x)) = R(n+1,x). It follows that the zeros of R(n,x) are real and negative (apply Corollary 1.2 of [Liu and Wang]).
Since this is the triangle of fvectors of the (simplicial complexes dual to the) type A permutohedra, whose hvectors form the Eulerian number triangle A008292, the coefficients of the polynomial (x1)^n*R(n,1/(x1) give the nth row of A008292. For example, from row 3 we have x^2 + 6*x + 6 = 1 + 4*y + y^2, where y = x + 1, producing [1,4,1] as the third row of A008292. The matrix product A008292 * A007318 gives the mirror image of this triangle (see A090582).
For n,k >= 0, T(n+1,k+1) = Sum_{j=0..k} (1)^(kj)*binomial(k,j)*[(j+1)^(n+1)  j^(n+1)]. The matrix product of Pascal's triangle A007318 with the current array gives (essentially) A047969. This triangle is also related to triangle A047969 by means of the Stransform of [Hetyei], a linear transformation of polynomials whose value on the basis monomials x^k is given by S(x^k) = binomial(x,k). The Stransform of the shifted nth row polynomial Q(n,x) := R(n,x)/x is S(Q(n,x)) = (x+1)^n  x^n. For example, from row 3 we obtain S(1 + 6*x + 6*x^2) = 1 + 6*x + 6*x*(x1)/2 = 1 + 3*x + 3*x^2 = (x+1)^3  x^3. For fixed k, the values S(Q(n,k)) give the nonzero entries in column (k1) of the triangle A047969 (the Hilbert transform of the Eulerian numbers). (End)
E.g.f.: (exp(x)1)^k = sum T(n,k)x^n/n!.  Vladimir Kruchinin, Aug 10 2010
T(n,k) = Sum_{i=1...k} A(n,i)*Binomial(ni,ki) where A(n,i) is the number of npermutations that have i ascending runs, A008292.
From Tom Copeland, Oct 11 2011: (Start)
With e.g.f.. A(x,t)= 1 + 1/(1+t*(1exp(x)), the comp. inverse in x is
B(x,t)= log(((1+t)/t)  1/(t(1+x))).
With h(x,t)= 1/(dB/dx)= (1+x)((1+t)(1+x)1), the row polynomial P(n,t) is given by (h(x,t)*d/dx)^n x, eval. at x=0, A=exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t), with P(0,t)=0.
(A factor of 1/n! was removed by Copeland on Aug 25 2016). (End)
The term linear in x of [x*h(d/dx,t)]^n 1 gives the nth row polynomial. (See A134685.)  Tom Copeland, Nov 07 2011
Row polynomials are given by D^n(1/(1x*t)) evaluated at x = 0, where D is the operator (1+x)*d/dx.  Peter Bala, Nov 25 2011
T(n,x+y) = Sum_{j=0..n} binomial(n,j)*T(j,x)*T(nj,y).  Dennis P. Walsh, Feb 24 2012
Let P be a RotaBaxter operator of weight 1 satisfying the identity P(x)*P(y) = P(P(x)*y) + P(x*P(y)) + P(x*y). Then P(1)^2 = P(1) + 2*P^2(1). More generally, Guo shows that P(1)^n = Sum_{k=1..n} T(n,k)*P^k(1).  Peter Bala, Jun 08 2012
Sum_{i=1..n} (1)^i*T(n,i)/i = 0, for n>1.  Leonid Bedratyuk, Aug 09 2012
T(n, k) = Sum_{j=0..k} (1)^j*binomial(k, j)*(kj)^n. [M. Catalani's reindexed formula from Nov 28 2003] Proof: count the number of surjections of [n] onto [k] with the inclusionexclusion principle, as an alternating sum of the number of functions from [n] to [kj].  Bert Seghers, Jun 29 2013
nth row polynomial = 1/(1 + x)*( Sum_{k>=0} k^n*(x/(1 + x))^k ), valid for x in the open interval (1/2, inf). See Tanny link. Cf. A145901.  Peter Bala, Jul 22 2014
T(n,k) = k * A141618(n,k1) / binomial(n,k1).  Tom Copeland, Oct 25 2014
Sum_{n=0..infinity} n^k*a^n = Sum_{i=1..k} (a / (1  a))^i * T(k, i)/(1a) for a<1.  David A. Corneth, Mar 09 2015
From Peter Bala, May 26 2015: (Start)
The row polynomials R(n,x) satisfy (1 + x)*R(n,x) = (1)^n*x*R(n,(1 + x)).
For a fixed integer k, the expansion of the function A(k,z) := exp( Sum_{n >= 1} R(n,k)*z^n/n ) has integer coefficients and satisfies the functional equation A(k,z)^(k + 1) = BINOMIAL(A(k,z))^k, where BINOMIAL(F(z))= 1/(1  z)*F(z/(1  z)) denotes the binomial transform of the o.g.f. F(z). Cf. A145901. For cases see A084784 (k = 1), A090352 (k = 2), A090355 (k = 3), A090357 (k = 4), A090362 (k = 5) and A084785 (k = 2 with z > z).
A(k,z)^(k + 1) = A((k + 1),z)^k and hence BINOMIAL(A(k,z)) = A((k + 1),z). (End)
From Tom Copeland, Oct 19 2016: (Start)
Let a(1) = 1 + x + B(1) = x + 1/2 and a(n) = B(n) = (B.)^n, where B(n) are the Bernoulli numbers defined by e^(B.t) = t / (e^t1), then t / e^(a.t) = t / [(x + 1) * t + exp(B.t)] = (e^t  1) /[ 1 + (x + 1) (e^t  1)] = exp(p.(x)t), where (p.(x))^n = p_n(x) are the shifted, signed row polynomials of this array: p_0(x) = 0, p_1(x) = 1, p_2(x) = (1 + 2 x), p_3(x) = 1 + 6 x + 6 x^2, ... and p_n(x) = n * b(n1), where b(n) are the partition polynomials of A133314 evaluated with these a(n).
Sum_{n > 0} R(n,1/2) x^n/n! = 2 * tanh(x/2), where R(n,x) = sum_{k = 1,..,n} T(n,k) x^(k1) are the shifted row polynomials of this entry, so R(n,1/2) = 4 * (2^(n+1)1) B(n+1)/(n+1). (Cf. A000182.)
(End)
Also the Bernoulli numbers are given by B(n) = Sum_{k =1..n} (1)^k T(n,k) / (k+1).  Tom Copeland, Nov 06 2016


EXAMPLE

The triangle T(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9 10
1: 1
2: 1 2
3: 1 6 6
4: 1 14 36 24
5: 1 30 150 240 120
6: 1 62 540 1560 1800 720
7: 1 126 1806 8400 16800 15120 5040
8: 1 254 5796 40824 126000 191520 141120 40320
9: 1 510 18150 186480 834120 1905120 2328480 1451520 362880
10: 1 1022 55980 818520 5103000 16435440 29635200 30240000 16329600 3628800
... Reformatted and extended  Wolfdieter Lang, Oct 04 2014

T(4,1) = 1: {1234}. T(4,2) = 14: {1}{234} (4 ways), {12}{34} (6 ways), {123}{4} (4 ways). T(4,3) = 36: {12}{3}{4} (12 ways), {1}{23}{4} (12 ways), {1}{2}{34} (12 ways). T(4,4) = 1: {1}{2}{3}{4} (1 way).


MAPLE

with(combinat): A019538 := (n, k)>k!*stirling2(n, k);


MATHEMATICA

Table[k! StirlingS2[n, k], {n, 9}, {k, n}] // Flatten


PROG

(PARI) {T(n, k) = if( k<0  k>n, 0, sum(i=0, k, (1)^i * binomial(k, i) * (ki)^n))}; /* Michael Somos, Oct 08 2003 */
(Haskell)
a019538 n k = a019538_tabl !! (n1) !! (k1)
a019538_row n = a019538_tabl !! (n1)
a019538_tabl = iterate f [1] where
f xs = zipWith (*) [1..] $ zipWith (+) ([0] ++ xs) (xs ++ [0])
 Reinhard Zumkeller, Dec 15 2013
(Sage) T(n, k) = factorial(k)*stirling_number2(n, k) # Danny Rorabaugh, Oct 10 2015


CROSSREFS

Row sums give A000670. Maximal terms in rows give A002869. Central terms T(2k1,k) give A233734.
Diagonal is n! (A000142). 2nd diagonal is A001286. 3rd diagonal is A037960.
Cf. A000918, A000919, A001117, A001118, A008275, A008279, A048594, A059117, A059515, A074909, A084938.
Reflected version of A090582.
See also the two closely related triangles: A008277(n, k) = T(n, k)/k! (Stirling numbers of second kind) and A028246(n, k) = T(n, k)/k.
Cf. A033282 'faces' of the associahedron.
Cf. A008292, A047969, A145901, A145902.  Peter Bala, Oct 26 2008
Visible in the 3D array in A249042.
Cf. A000182.
Sequence in context: A202190 A089231 A052296 * A269646 A269336 A046521
Adjacent sequences: A019535 A019536 A019537 * A019539 A019540 A019541


KEYWORD

nonn,tabl,easy,nice


AUTHOR

N. J. A. Sloane, Manfred Goebel (goebel(AT)informatik.unituebingen.de), Dec 11 1996


EXTENSIONS

Boole reference from Michael Somos, Oct 10 2003


STATUS

approved



