Search: id:A019538 Results 1-1 of 1 results found. %I A019538 %S A019538 1,1,2,1,6,6,1,14,36,24,1,30,150,240,120,1,62,540,1560,1800,720,1,126, %T A019538 1806,8400,16800,15120,5040,1,254,5796,40824,126000,191520,141120,40320, %U A019538 1,510,18150,186480,834120,1905120,2328480,1451520,362880,1,1022,55980 %N A019538 Triangle of numbers T(n,k) = k!*Stirling2(n,k) read by rows (n >= 1, 1 <= k <= n). %C A019538 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. %C A019538 Number of onto functions from an n-element set to a k-element set. %C A019538 Also coefficients (in ascending order) of so-called ordered Bell polynomials. %C A019538 (k-1)!*Stirling2(n,k-1) is the number of chain topologies on an n-set having k open sets [Stephen]. %C A019538 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 %C A019538 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] %C A019538 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] %C A019538 Subtriangle of triangle in A131689. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 03 2008] %D A019538 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. %D A019538 Moussa Benoumhani, The number of topologies on a finite set, Preprint, 2005. %D A019538 G. Boole, A Treatise On The Calculus of Finite Differences, Dover, 1960, p. 20. %D A019538 J. L. Chandon, J. LeMaire and J. Pouget, Denombrement des quasi-ordres sur un ensemble fini, Math. Sci. Humaines, No. 62 (1978), 61-80. %D A019538 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. %D A019538 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] %D A019538 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] %D A019538 G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30. %D A019538 E. Mendelsohn, Races with ties, Math. Mag. 55 (1982), 170-175. %D A019538 T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. %D A019538 T. S. Motzkin, Sorting numbers ...: for a link to this paper see A000262. %D A019538 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 33. %D A019538 R. Simion, "Convex Polytopes and Enumeration", Adv. in Appl. Math. 18 (1997) pp. 149-180. %D A019538 J. F. Steffensen, Interpolation, 2nd ed., Chelsea, NY, 1950, see p. 54. %D A019538 D. Stephen, Topology on finite sets, Amer. Math. Monthly, 75 (1968), 739-741. %D A019538 A. H. Voigt, Theorie der Zahlenreihen und der Reihengleichungen, Goschen, Leipzig, 1911, p. 31. %D A019538 E. Whittaker and G. Robinson, The Calculus of Observations, Blackie, London, 4-th ed., 1949; p. 7. %H A019538 T. D. Noe, Rows n=1..100 of triangle, flattened %H A019538 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] %H A019538 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.) %H A019538 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] %H A019538 L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008] %H A019538 J. Loday, The Multiple Facets of the Associahedron [From Tom Copeland (tcjpn(AT)msn.com), Sep 29 2008] %H A019538 K. A. Penson, P. Blasiak, G. Duchamp, A. Horzela and A. I. Solomon, Hierarchical Dobinski-type relations via substitution and the moment problem %H A019538 Wikipedia, Truncated octahedron [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008] %F A019538 T(n, k) = Sum_{j=0..k} (-1)^j*C(k, j)*(k-j)^n. %F A019538 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 %F A019538 E.g.f.: (y*(exp(x)-1)-exp(x))/(y*(exp(x)-1)-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 30 2003 %F A019538 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. %F A019538 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 %F A019538 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 %F A019538 O.g.f. for n-th row: polylog(-n, x/(1+x))/(x+x^2). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 30 2005 %F A019538 E.g.f. 1 / {1 + t[1-exp(x)]} [From Tom Copeland (tcjpn(AT)msn.com), Oct 13 2008] %F A019538 Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008: (Start) %F A019538 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 + ... . %F A019538 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]). %F A019538 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). %F A019538 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) %e A019538 Triangle begins: %e A019538 1 %e A019538 1,2 %e A019538 1,6,6 %e A019538 1,14,36,24 %e A019538 1,30,150,240,120 %e A019538 ... %e A019538 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). %p A019538 with(combinat): A019538 := (n,k)->k!*stirling2(n,k); %o A019538 (PARI) T(n,k)=if(k<0|k>n,0,sum(i=0,k,(-1)^i*binomial(k,i)*(k-i)^n)) %Y A019538 Row sums give A000670. 2nd diagonal is A001286. 3rd diag. is A037960. Maximal terms in rows give A002869. Cf. A008275, A048594. %Y A019538 Cf. A008277, A000918, A001117, A000919, A001118, A059117, A059515, A084938. %Y A019538 Reflected version of A090582. Diagonal is n! (A000142). %Y A019538 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. %Y A019538 Cf. A033282 'faces' of the associahedron. %Y A019538 Cf. A008292, A047969, A145901, A145902. [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008] %Y A019538 Sequence in context: A063007 A089231 A052296 this_sequence A046521 A104684 A060538 %Y A019538 Adjacent sequences: A019535 A019536 A019537 this_sequence A019539 A019540 A019541 %K A019538 nonn,tabl,easy,nice %O A019538 1,3 %A A019538 N. J. A. Sloane (njas(AT)research.att.com), Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de) %E A019538 Boole reference from Michael Somos, Oct 10 2003 Search completed in 0.002 seconds