A004123 Number of generalized weak orders on n points.

1, 2, 10, 74, 730, 9002, 133210, 2299754, 45375130, 1007179562, 24840104410, 673895590634, 19944372341530, 639455369290922, 22079273878443610, 816812844197444714, 32232133532123179930, 1351401783010933015082

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy

Comments

Number of bipartitional relations on a set of cardinality n. - Ralf Stephan, Apr 27 2003

From Peter Bala, Jul 08 2022: (Start)

Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 2, 3, 4, 2, 0, 0, 2, 3, 4, 2, 0, 0, 2, 3, 4, 2, 0, 0, ...] with an apparent period of 6 = phi(7) starting at a(2). Cf. A000670.

More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series. (End)

References

• L Santocanale, F Wehrung, G Grätzer, F Wehrung, Generalizations of the Permutohedron, in Grätzer G., Wehrung F. (eds) Lattice Theory: Special Topics and Applications. Birkhäuser, Cham, pp. 287-397; DOI https://doi.org/10.1007/978-3-319-44236-5_8
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 1..100
• Paul Barry, Eulerian-Dowling Polynomials as Moments, Using Riordan Arrays, arXiv:1702.04007 [math.CO], 2017.
• Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
• P. Blasiak, K. A. Penson and A. I. Solomon, Dobinski-type relations and the log-normal distribution. arXiv:quant-ph/0303030, 2003.
• C. G. Bower, Transforms
• D. Foata and C. Krattenthaler, Graphical Major Indices, II, Seminaire Lotharingien de Combinatoire, B34k, 16 pp., 1995.
• D. Foata and D. Zeilberger, The Graphical Major Index, arXiv:math/9406220 [math.CO], 1994.
• Jacob Sprittulla, On Colored Factorizations, arXiv:2008.09984 [math.CO], 2020.
• Carl G. Wagner, Enumeration of generalized weak orders, Arch. Math. (Basel) 39 (1982), no. 2, 147-152.
• C. G. Wagner, Enumeration of generalized weak orders, Preprint, 1980. [Annotated scanned copy]
• C. G. Wagner and N. J. A. Sloane, Correspondence, 1980

Crossrefs

Cf. A004121, A004122, A000165, A000670, A032033.

Second row of array A094416 (generalized ordered Bell numbers).

Equals 2 * A050351(n) for n>0.

Programs

Mathematica

a[n_] := (1/3)*PolyLog[-n + 1, 2/3]; a[1]=1; Table[a[n], {n, 1, 18}] (* Jean-François Alcover, Jun 11 2012 *)
CoefficientList[Series[1/(3-2*Exp[x]), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Aug 07 2013 *)

PARI

{a(n)=polcoeff(sum(m=0, n, 2^m*m!*x^(m+1)/prod(k=1, m, 1-k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */

PARI

my(N=25,x='x+O('x^N)); Vec(serlaplace(1/(3 - 2*exp(x)))) \\ Joerg Arndt, Jan 15 2024

Sage

A004123 = lambda n: sum(stirling_number2(n-1,k)*(2^k)*factorial(k) for k in (0..n-1))
[A004123(n) for n in (1..18)] # Peter Luschny, Jan 18 2016

Formula

E.g.f. for sequence with offset 0: 1/(3-2*exp(x)).

a(n) = 2^n*A(n,3/2); A(n,x) the Eulerian polynomials. - Peter Luschny, Aug 03 2010

O.g.f.: Sum_{n>=0} 2^n*n!*x^(n+1)/Product_{k=0..n} (1-k*x). - Paul D. Hanna, Jul 20 2011

a(n) = Sum_{k>=0} k^n*(2/3)^k/3.

a(n) = Sum_{k=0..n} Stirling2(n, k)*(2^k)*k!.

Stirling transform of A000165. - Karol A. Penson, Jan 25 2002

"AIJ" (ordered, indistinct, labeled) transform of 2, 2, 2, 2, ...

Recurrence: a(n) = 2*Sum_{k=1..n} binomial(n, k)*a(n-k), a(0)=1. - Vladeta Jovovic, Mar 27 2003

a(n) ~ (n-1)!/(3*(log(3/2))^n). - Vaclav Kotesovec, Aug 07 2013

a(n) = log(3/2)*Integral_{x>=0} floor(x)^n * (3/2)^(-x) dx. - Peter Bala, Feb 14 2015

E.g.f.: (x - log(3 - 2*exp(x)))/3. - Ilya Gutkovskiy, May 31 2018

Conjectural o.g.f. as a continued fraction of Stieltjes type: 1/(1 - 2*x/(1 - 3*x/(1 - 4*x/(1 - 6*x/(1 - ... - 2*n*x/(1 - 3*n*x/(1 - ...))))))). - Peter Bala, Jul 08 2022

Extensions

More terms from Christian G. Bower