%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, <a href="b019538.txt">Rows n=1..100 of triangle, flattened</
               a>
%H A019538 S. Fomin, N. Reading, <a href="http://arxiv.org/abs/math.CO/0505518">
               Root systems and generalized associahedra</a>, Lecture notes for 
               IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct 
               26 2008]
%H A019538 M. Goebel, <a href="http://www.informatik.uni-trier.de/~ley/db/journals/
               aaecc/aaecc8.html">On the number of special permutation-invariant 
               orbits and terms</a>, 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, <a href="http://www.math.uncc.edu/preprint/2004/2004_11.pdf">
               Face enumeration using generalized binomial coefficients</a>. 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 href="http://www.arXiv.org/abs/math.CO/0509207">A 
               unified approach to polynomial sequences with only real zeros</a> 
               [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]
%H A019538 J. Loday, <a href="http://www.claymath.org/programs/outreach/academy/
               LectureNotes05/Lodaypaper.pdf">The Multiple Facets of the Associahedron</
               a> [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, <a 
               href="http://arXiv.org/abs/quant-ph/0312202">Hierarchical Dobinski-type 
               relations via substitution and the moment problem</a>
%H A019538 Wikipedia, <a href="http://en.wikipedia.org/wiki/Truncated_octahedron">
               Truncated octahedron</a> [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