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

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

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.

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.

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

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

D. Stephen, Toplogy 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.

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

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

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.)

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

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.yu), 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.yu), Jan 30 2005

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.

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

njas, Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de)

Boole reference from Michael Somos, Oct 10 2003