1,3

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. Also called differences of 0.

Number of onto functions from an n-element set to a k-element set.

Also coefficients (in ascending order) of so-called ordered Bell polynomials.

(k-1)!*Stirling2(n,k-1) is the number of chain topologies on an n-set 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 (maharri(AT)gmail.com), Jan 16 2007

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 (0-D faces), 36 edges (1-D faces), 6 squares (2-D faces), 8 hexagons (2-D faces) and 1 3-D permutohedron. Summing coefficients over like dimensions gives A019538 and A090582. Compare to A133437 for the associahedron. [From Tom Copeland (tcjpn(AT)msn.com), Sep 29 2008, Oct 07 2008]

Further to the comments of T. 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 f-vector (1,14,36,24) giving the fourth row of this triangle. See the Wikipedia entry and [Fomin and Reading p.21]. The corresponding h-vectors of permutohedra of type A give the rows of the triangle of Eulerian numbers A008292. See A145901 and A145902 for the array of f-vectors for type B and type D permutohedra respectively. [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]

Subtriangle of triangle in A131689. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2008]

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.

G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 20.

J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80.

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.

Gabor Hetyei. The Stirling polynomial of a simplicial complex. Discrete and Computational Geometry 35, Number 3, March 2006, pp 437-455. [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]

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 [From N. J. A. Sloane, Jul 10 2009]

G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.

E. Mendelsohn, Races with ties, Math. Mag. 55 (1982), 170-175.

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.

T. S. Motzkin, Sorting numbers ...: for a link to this paper see A000262.

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33.

R. Simion, "Convex Polytopes and Enumeration", Adv. in Appl. Math. 18 (1997) pp. 149-180.

J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54.

D. Stephen, Topology on finite sets, Amer. Math. Monthly, 75 (1968), 739-741.

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, 4-th ed., 1949; p. 7.

T. D. Noe, Rows n=1..100 of triangle, flattened

S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]

M. Goebel, On the number of special permutation-invariant orbits and terms, in Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.)

G. Hetyei, Face enumeration using generalized binomial coefficients. Online draft version of the Hetyei paper referenced above. [From Peter Bala (pbala(AT)toucansurf.com), Nov 10 2008]

L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]

J. Loday, The Multiple Facets of the Associahedron [From Tom Copeland (tcjpn(AT)msn.com), Sep 29 2008]

K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem

Wikipedia, Truncated octahedron [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]

T(n, k) = Sum_{j=0..k} (-1)^j*C(k, j)*(k-j)^n.

T(n, k) = k*(T(n-1, k-1)+T(n-1, k)) with T(0, 0) = 1 [or T(1, 1) = 1] - Henry Bottomley (se16(AT)btinternet.com), Mar 02 2001

E.g.f.: (y*(exp(x)-1)-exp(x))/(y*(exp(x)-1)-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), 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 Deleham's operator defined in A084938.

Also T(n, k)=Sum((-1)^(k-j)j^n*C(k, j), j=0, .., k) - Mario Catalani (mario.catalani(AT)unito.it), Nov 28 2003

Sum(T(n, k)(-1)^(n-k), k=0, .., n)=1, Sum(T(n, k)(-1)^k, k=0, .., n)=(-1)^n. - Mario Catalani (mario.catalani(AT)unito.it), Dec 11 2003

O.g.f. for n-th row: polylog(-n, x/(1+x))/(x+x^2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 30 2005

E.g.f. 1 / {1 + t[1-exp(x)]} [From Tom Copeland (tcjpn(AT)msn.com), Oct 13 2008]

Contribution from Peter Bala (pbala(AT)toucansurf.com), 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 f-vectors of the (simplicial complexes dual to the) type A permutohedra, whose h-vectors form the Eulerian number triangle A008292, the coefficients of the polynomial (x-1)^n*R(n,1/(x-1) give the n-th 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)^(k-j)*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 S-transform 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 S-transform of the shifted n-th 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*(x-1)/2 = 1 + 3*x + 3*x^2 = (x+1)^3 - x^3. For fixed k, the values S(Q(n,k)) give the non-zero entries in column (k-1) of the triangle A047969 (the Hilbert transform of the Eulerian numbers). (End)

Triangle begins:

1

1,2

1,6,6

1,14,36,24

1,30,150,240,120

...

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).

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

(PARI) T(n, k)=if(k<0|k>n, 0, sum(i=0, k, (-1)^i*binomial(k, i)*(k-i)^n))

Row sums give A000670. 2nd diagonal is A001286. 3rd diag. is A037960. Maximal terms in rows give A002869. Cf. A008275, A048594.

Cf. A008277, A000918, A001117, A000919, A001118, A059117, A059515, A084938.

Reflected version of A090582. Diagonal is n! (A000142).

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. [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]

Sequence in context: A063007 A089231 A052296 this_sequence A046521 A104684 A060538

Adjacent sequences: A019535 A019536 A019537 this_sequence A019539 A019540 A019541

nonn,tabl,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de)

Boole reference from Michael Somos, Oct 10 2003