A052296 -id:A052296 - cp's OEIS frontend

A001831 Number of labeled graded partially ordered sets with n elements of height at most 1.

1, 1, 3, 13, 87, 841, 11643, 227893, 6285807, 243593041, 13262556723, 1014466283293, 109128015915207, 16521353903210521, 3524056001906654763, 1059868947134489801413, 449831067019305308555487, 269568708630308018001547681, 228228540531327778410439620963

Offset: 0

N. J. A. Sloane

Labeled posets where for all a,b,c in the set, do not have a

Number of labeled digraphs with n vertices with no directed path of length 2. Number of n X n {0,1} matrices A such that A^2 = 0. - Michael Somos, Jul 28 2013

Number of relations on n labeled nodes that are simultaneously transitive and antitransitive. - Peter Kagey, Feb 14 2021

			1 + x + 3*x^2 + 13*x^3 + 87*x^4 + 841*x^5 + 11643*x^6 + 227893*x^7 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 0..50
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19. [Annotated scanned copy]
D. A. Klarner, The number of graded partially ordered sets, J. Combin. Theory, 6 (1969), 12-19.
A. Motzek and R. Möller, Exploiting Innocuousness in Bayesian Networks, Preprint, Australasian Joint Conference on Artificial Intelligence AI 2015: Advances in Artificial Intelligence, pp. 411-423.
Zvi Rosen and Yan X. Zhang, Convex Neural Codes in Dimension 1, arXiv:1702.06907 [math.CO], 2017. Mentions this sequence.
Index entries for sequences related to posets

Cf. A002031, A047863, A052332, A001833, A048194.
Row sums of A052296.
Cf. variants: A135753, A135754.

A001831 := proc(n)
    add(binomial(n,k)*(2^k-1)^(n-k),k=0..n) ;
end proc:
seq(A001831(n),n=0..10) ; # R. J. Mathar, Mar 08 2021

Join[{1}, Table[Sum[Binomial[n,k](2^k-1)^(n-k),{k,n}],{n,20}]] (* Harvey P. Dale, Jan 05 2012 *)

{a(n)=n!*polcoeff(sum(k=0,n,exp((2^k-1)*x)*x^k/k!),n)} \\ Paul D. Hanna, Nov 27 2007

{a(n)=polcoeff(sum(k=0, n, x^k/(1-(2^k-1)*x +x*O(x^n))^(k+1)), n)} \\ Paul D. Hanna, Sep 15 2009

a(n) = Sum((-1)^k*C(n, k)*A047863(k), k=0..n).
a(n) = Sum_{k=0..n} binomial(n, k)*(2^k-1)^(n-k). - Vladeta Jovovic, Apr 04 2003
E.g.f.: Sum_{n>=0} exp((2^n-1)*x) * x^n/n!. - Paul D. Hanna, Nov 27 2007 [correction made by Paul D. Hanna, Mar 08 2021]
O.g.f.: Sum_{n>=0} x^n/(1 - (2^n - 1)*x)^(n+1) = Sum_{n>=0} a(n)*x^n. - Paul D. Hanna, Sep 15 2009
a(n) ~ c * 2^(n^2/4 + n + 1/2) / sqrt(Pi*n), where c = JacobiTheta3(0,1/2) = EllipticTheta[3, 0, 1/2] = 2.1289368272118771586694585485449... if n is even, and c = JacobiTheta2(0,1/2) = EllipticTheta[2, 0, 1/2] = 2.1289312505130275585916134025753... if n is odd. - Vaclav Kotesovec, Mar 10 2014

More terms, formula and comments from Christian G. Bower, Dec 15 1999
Last 4 terms corrected by Vladeta Jovovic, Apr 04 2003
Comments corrected by Joel B. Lewis, Mar 28 2011

A117279 Triangle read by rows: T(n,k) is number of labeled bipartite graphs with n nodes and k edges.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 3, 1, 6, 15, 16, 3, 1, 10, 45, 110, 140, 60, 10, 1, 15, 105, 435, 1125, 1701, 1200, 480, 105, 10, 1, 21, 210, 1295, 5355, 14952, 26572, 26670, 17535, 7840, 2331, 420, 35, 1, 28, 378, 3220, 19075, 81228, 246414, 507424, 666015, 620900, 431368

Offset: 0

Vladeta Jovovic, Jun 23 2007

			Triangle begins:
  1;
  1;
  1,  1;
  1,  3,  3;
  1,  6, 15,  16,   3;
  1, 10, 45, 110, 140, 60, 10;
  ...

R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.5.

Andrew Howroyd, Table of n, a(n) for n = 0..1403 (rows 0..25)

Row sums give A047864,
Columns k=1..5 are A000217(n-1), A050534, A053526, A053527, A053528.
The unlabeled version is A297877.
Cf. A052296, A033995.

nn=10;f[x_,y_]:=Sum[Sum[Binomial[n,k](1+y)^(k(n-k)),{k,0,n}]x^n/n!,{n,0,nn}];Map[Select[#,#>0&]&,Range[0,nn]!CoefficientList[Series[Exp[Log[f[x,y]]/2],{x,0,nn}],{x,y}]]//Grid (* Geoffrey Critzer, Sep 05 2013 *)

T(n)={[Vecrev(p) | p<-Vec(serlaplace(sqrt(sum(k=0, n, exp(x*(1+y)^k + O(x*x^n))*x^k/k! ))))]}
{ my(A=T(6)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jan 10 2022

E.g.f.: sqrt(Sum_{n>=0} exp(x*(1+q)^n)*x^n/n!).

A055531 Number of labeled order relations on n nodes in which longest chain has 2 nodes.

Original entry on oeis.org

2, 12, 86, 840, 11642, 227892, 6285806, 243593040, 13262556722, 1014466283292, 109128015915206, 16521353903210520, 3524056001906654762, 1059868947134489801412, 449831067019305308555486, 269568708630308018001547680, 228228540531327778410439620962

Offset: 2

N. J. A. Sloane, Jul 10 2000

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 60.

A column or diagonal of triangle in A342587.

Cf. A001831, A052296.

a(n) = A001831(n)-1.

a(n) = Sum_{k=1..n-1} binomial(n,k)*(2^(n-k)-1)^k. - Geoffrey Critzer, Oct 29 2023

More terms from Vladeta Jovovic, Oct 24 2000

a(13)-a(16) corrected and more terms from Sean A. Irvine, Mar 25 2022

A369921 Number of cover relations summed over the rank-1 labeled posets on [n].

Original entry on oeis.org

0, 0, 2, 18, 204, 2940, 56670, 1471806, 52067512, 2520298584, 167850357210, 15435027907530, 1967345286257604, 348527628228821652, 86057693880611800438, 29677160119074814383030, 14321851348104417100842480

Offset: 0

Geoffrey Critzer, Feb 05 2024

nonn

The rank of a poset is the number of cover relations in a maximal chain.

A cover relation in a poset is an ordered pair x <= y such that there is no z with x <= z <= y.

Eric Weisstein's World of Mathematics, Cover Relation.

Cf. A001831, A369919, A052296.

nn = 16; Table[Table[n!, {n, 0, nn}] CoefficientList[D[Series[Sum[Exp[y x]^Binomial[n, i]*Exp[  x]^(2^n - Binomial[n, i] - 1)  x^n/n!, {n, 0, nn}], {x, 0, nn}], y] /. y -> 1, x]*i, {i, 1, nn - 1}] // Total

a(n) = Sum_{k=1..floor(n^2/4)} A052296(n,k)*k.

A174122 Partial sums of A001831.

Original entry on oeis.org

1, 2, 5, 18, 105, 946, 12589, 240482, 6526289, 250119330, 13512676053, 1027978959346, 110155994874553, 16631509898085074, 3540687511804739837, 1063409634646294541250, 450894476653951603096737

Offset: 0

Jonathan Vos Post, Mar 08 2010

nonn

Partial sums of number of labeled graded partially ordered sets with n elements. The subsequence of primes in this partial sum begins: 2, 5, 12589.

Cf. A001831, A002031, A047863, A052332, A001833, A048194, A052296, A135753, A135754.

a(n) = SUM[i=0..n] A001831(i) = SUM[i=0..n] SUM[j=0..i] ((-1)^j*C(n,j)*A047863(j)).

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A001831 Number of labeled graded partially ordered sets with n elements of height at most 1.

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A117279 Triangle read by rows: T(n,k) is number of labeled bipartite graphs with n nodes and k edges.

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A055531 Number of labeled order relations on n nodes in which longest chain has 2 nodes.

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A369921 Number of cover relations summed over the rank-1 labeled posets on [n].

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A174122 Partial sums of A001831.

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