A048194 -id:A048194 - 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

A048192 Number of connected chordal graphs on n vertices.

Original entry on oeis.org

1, 1, 2, 5, 15, 58, 272, 1614, 11911, 109539, 1247691, 17566431, 305310547, 6558690953, 174688164414

Offset: 1

Gordon F. Royle

F. Hüffner, tinygraph, software for generating integer sequences based on graph properties, version 6e0a59d
Tim Kemp, An algorithmic approach to a conjecture of Chvátal on toughness and hamiltonicity of graphs, M. Sc. Thesis, University of Twente (Netherlands, 2020).
B. D. McKay, Chordal graphs
Eric Weisstein's World of Mathematics, Chordal Graph
Eric Weisstein's World of Mathematics, Connected Graph

Cf. A048193 (not-necessarily connected chordal graphs).
Cf. A287427 (disconnected chordal graphs).
Cf. A048194.

a(n) = A048193(n) - A287427(n). - Eric W. Weisstein, May 25 2017

Inverse Euler transform of A048193. - Andrew Howroyd, Nov 03 2017

a(12) added by Gordon F. Royle, Aug 05 2008
a(13) and a(14) added using tinygraph by Falk Hüffner, Jan 15 2016
a(15) added by Brendan McKay, Jan 07 2019

A055080 Triangle T(n,k) read by rows, giving number of k-member minimal covers of an unlabeled n-set, k=1..n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 9, 4, 1, 1, 9, 23, 17, 5, 1, 1, 12, 51, 65, 28, 6, 1, 1, 16, 103, 230, 156, 43, 7, 1, 1, 20, 196, 736, 863, 336, 62, 8, 1, 1, 25, 348, 2197, 4571, 2864, 664, 86, 9, 1, 1, 30, 590, 6093, 22952, 25326, 8609, 1229, 115, 10, 1, 1, 36, 960

Offset: 1

Vladeta Jovovic, Jun 13 2000

Also number of unlabeled split graphs on n vertices and with a k-element clique (cf. A048194).

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   3,   1;
  1,  6,   9,   4,   1;
  1,  9,  23,  17,   5,   1;
  1, 12,  51,  65,  28,   6,  1;
  1, 16, 103, 230, 156,  43,  7, 1;
  1, 20, 196, 736, 863, 336, 62, 8, 1;
  ...
There are four minimal covers of an unlabeled 3-set: one 1-cover {{1,2,3}}, two 2-covers {{1,2},{3}}, {{1,2},{1,3}} and one 3-cover {{1},{2},{3}}.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
R. J. Clarke, Covering a set by subsets, Discrete Math., 81 (1990), 147-152.
Vladeta Jovovic, Binary matrices up to row and column permutations
G. F. Royle, Counting Set Covers and Split Graphs, J. Integer Seqs., 3 (2000), #00.2.6.
Eric Weisstein's World of Mathematics, Minimal covers

Row sums give A048194.
Columns are A000012, A087811, A005783, A005784, A005785 A005786, A055066.
Diagonals are A000012, A000027, A005744, A005745, A005746, A005771, A005747, A005748.
Cf. A035348 for labeled case.
Cf. A002620, A028657.

\\ Needs A(n,m) from A028657.
T(n,k) = A(n-k, k) - if(kAndrew Howroyd, Feb 28 2023

T(n,k) = A028657(n,k) - A028657(n-1,k). - Andrew Howroyd, Feb 28 2023

A049312 Number of graphs with a distinguished bipartite block, by number of vertices.

Original entry on oeis.org

1, 2, 4, 8, 17, 38, 94, 258, 815, 3038, 13804, 78760, 580456, 5647602, 73645352, 1297920850, 31031370360, 1007551636038, 44432872400460, 2661065508648436, 216457998880015366, 23920728651724212120, 3593384834863975164882, 734240676501745813835934

Offset: 0

Peter J. Cameron

Calculate number of connected bipartite graphs + number of connected bipartite graphs with no duality automorphism, apply EULER transform.

Inverse Euler transform is A318870.

			a(2)=4: null graph with 0, 1 or 2 vertices in the distinguished block and complete graph with 1 vertex in distinguished block.

R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

Alois P. Heinz, Table of n, a(n) for n = 0..40
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Karen L. Collins, Ann N. Trenk, Finding Balance: Split Graphs and Related Classes, arXiv:1706.03092 [math.CO], June 2017.
M. Guay-Paquet, A. H. Morales and E. Rowland, Structure and enumeration of (3+ 1)-free posets, arXiv preprint arXiv:1212.5356 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Feb 01 2013
J. M. Troyka, Split graphs: combinatorial species and asymptotics, arXiv:1803.07248 [math.CO], 2018-2019.
J. M. Troyka, Split graphs: combinatorial species and asymptotics, Electron. J. Combin., 26 (2019), #P2.42.
E. M. Wright, The k-connectedness of bipartite graphs, J. Lond. Math. Soc. (2), 25 (1982), 7-12.

Row sums of A028657.

Cf. A048194, A318870.

b:= proc(n, i) option remember; `if`(n=0, {0}, `if`(i<1, {},
      {seq(map(p-> p+j*x^i, b(n-i*j, i-1) )[], j=0..n/i)}))
    end:
g:= proc(n, k) option remember; add(add(2^add(add(igcd(i, j)*
      coeff(s, x, i)* coeff(t, x, j), j=1..degree(t)),
      i=1..degree(s))/mul(i^coeff(s, x, i)*coeff(s, x, i)!,
      i=1..degree(s))/mul(i^coeff(t, x, i)*coeff(t, x, i)!,
      i=1..degree(t)), t=b(n+k$2)), s=b(n$2))
    end:
A:= (n, k)-> g(min(n, k), abs(n-k)):
a:= d-> add(A(n, d-n), n=0..d):
seq(a(n), n=0..20);  # Alois P. Heinz, Aug 01 2014

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i<1, {}, Flatten @ Table[ Map[ Function[ {p}, p+j*x^i], b[n-i*j, i-1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[ Sum[ 2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]*Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n+k, n+k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n-k]];
a[d_] := Sum[A[n, d-n], {n, 0, d}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Feb 25 2015, after Alois P. Heinz *)

a(n) ~ 1/n! A047863(n) = 1/n! Sum_{k=0..n} binomial(n,k) * 2^(k(n-k)) (see Wright; see also Thm. 3.7 of the Troyka link, which cites Wright). - Justin M. Troyka, Oct 29 2018

More terms from Vladeta Jovovic, Jun 17 2000

A056885 Triangle T(n,k) = number of minimal covers of an unlabeled n-set that cover k points of that set uniquely, k=0..n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 0, 0, 1, 3, 0, 0, 1, 3, 5, 0, 0, 1, 5, 8, 7, 0, 0, 1, 8, 19, 17, 11, 0, 0, 1, 12, 45, 56, 35, 15, 0, 0, 1, 17, 107, 194, 151, 65, 22, 0, 0, 1, 24, 244, 713, 728, 365, 118, 30, 0, 0, 1, 32, 547, 2697, 3996, 2413, 835, 203, 42, 0, 0, 1, 42, 1173, 10356, 24446

Offset: 0

Vladeta Jovovic, Sep 04 2000

Row sums give A048194.

			[1], [0,1], [0,0,2], [0,0,1,3], [0,0,1,3,5], [0,0,1,5,8,7], ...; There are 21=1+5+8+7 minimal covers of an unlabeled 5-set.

Sean A. Irvine, Java program (github)
V. Jovovic, Formula for T(n,k).

Cf. A035347 for labeled case.

A007776 Number of connected posets with n elements of height 1.

Original entry on oeis.org

1, 2, 4, 10, 27, 88, 328, 1460, 7799, 51196, 422521, 4483460, 62330116, 1150504224, 28434624153, 945480850638, 42417674401330, 2572198227615998, 211135833162079184, 23487811567341121158, 3545543330739039981738, 727053904070651775719646

Offset: 2

Georg Wambach (gw(AT)informatik.Uni-Koeln.de)

nonn

Inverse Euler transform of A048194 and A049312. - Detlef Pauly (dettodet(AT)yahoo.de) and Vladeta Jovovic, Jul 25 2003
Essentially the same as A318870. - Georg Fischer, Oct 02 2018
Number of connected digraphs on n unlabeled nodes where every node has indegree 0 or outdegree 0 and there are no isolated nodes.  - Andrew Howroyd, Oct 03 2018

Andrew Howroyd, Table of n, a(n) for n = 2..50 (terms 2..40 from Alois P. Heinz)
N. J. A. Sloane, Transforms
J. Textor, A. Idelberger, and M. Liskiewicz, Learning from Pairwise Marginal Independencies, arXiv:1508.00280 [cs.AI], 2015.
Index entries for sequences related to posets

Cf. A005142, A002031 (labeled case), A048194, A049312, A055192, A318870, column 1 of A342500.

mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i - 1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
b[d_] := Sum[A[n, d - n], {n, 0, d}];
EULERi[Array[b, 30]] // Rest (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz in A049312 *)

Inverse Euler transform of A055192. - Andrew Howroyd, Oct 03 2018

More terms from Vladeta Jovovic, Jul 25 2003

Offset corrected by Andrew Howroyd, Oct 03 2018

A055192 Number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block, up to isomorphism.

Original entry on oeis.org

1, 2, 5, 12, 35, 108, 393, 1666, 8543, 54190, 436740, 4565450, 62930604, 1156277748, 28509174012, 946786816168, 42448800498744, 2573207315483554, 211180300735118954, 23490473719472829824, 3545759835559406756008, 727077827560669587718290

Offset: 2

Vladeta Jovovic, Jun 18 2000

nonn

Also the number of connected split graphs on n vertices (cf. A048194). - Falk Hüffner, Dec 01 2015

Inverse Euler transform is A007776. - Andrew Howroyd, Oct 03 2018

Alois P. Heinz, Table of n, a(n) for n = 2..40

Equals second differences of A049312.
Cf. A007776, A024206, A055609, A055082, A055083, A055084.
Row sums of A056152 and also of A122083.

b[n_, i_] := b[n, i] = If[n == 0, {0}, If[i < 1, {}, Flatten @ Table[ Map[ Function[{p}, p + j*x^i], b[n - i*j, i - 1]], {j, 0, n/i}]]];
g[n_, k_] := g[n, k] = Sum[Sum[2^Sum[Sum[GCD[i, j]*Coefficient[s, x, i]* Coefficient[t, x, j], {j, 1, Exponent[t, x]}], {i, 1, Exponent[s, x]}]/ Product[i^Coefficient[s, x, i]*Coefficient[s, x, i]!, {i, 1, Exponent[s, x]}]/Product[i^Coefficient[t, x, i]*Coefficient[t, x, i]!, {i, 1, Exponent[t, x]}], {t, b[n + k, n + k]}], {s, b[n, n]}];
A[n_, k_] := g[Min[n, k], Abs[n - k]];
A049312[d_] := Sum[A[n, d - n], {n, 0, d}];
Differences[Table[A049312[n], {n, 0, 23}], 2] (* Jean-François Alcover, Sep 05 2019, after Alois P. Heinz in A049312 *)

A056152 Triangular array giving number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k=1..n-1 vertices, up to isomorphism.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 5, 5, 1, 1, 8, 17, 8, 1, 1, 11, 42, 42, 11, 1, 1, 15, 91, 179, 91, 15, 1, 1, 19, 180, 633, 633, 180, 19, 1, 1, 24, 328, 2001, 3835, 2001, 328, 24, 1, 1, 29, 565, 5745, 20755, 20755, 5745, 565, 29, 1, 1, 35, 930, 15274, 102089, 200082, 102089

Offset: 2

Vladeta Jovovic, Jul 29 2000

Also table read by rows: for 0 < k < n, a(n, k) = number of bipartite graphs with n vertices, no isolated vertices and a distinguished bipartite block with k vertices, up to isomorphism.
a(n, k) is the number of isomorphism classes of finite subdirectly irreducible almost distributive lattices in which the N-quotient has k upper covers and (n - k) lower covers. - David Wasserman, Feb 11 2002
Also, row n gives the number of unlabeled bicolored graphs having k nodes of one color and n-k nodes of the other color, with no isolated nodes; the color classes are not interchangeable.

			Triangle begins:
  1;
  1,  1;
  1,  3,   1;
  1,  5,   5,   1;
  1,  8,  17,   8,  1;
  1, 11,  42,  42,  11,  1;
  1, 15,  91, 179,  91,  15,  1;
  1, 19, 180, 633, 633, 180, 19, 1;
  ...
There are 17 bipartite graphs with 6 vertices, no isolated vertices and a distinguished bipartite block with 3 vertices, or equivalently, there are 17 3 X 3 binary matrices with no zero rows or columns, up to row and column permutation:
[0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1] [0 0 1]
[0 0 1] [0 0 1] [0 1 0] [0 1 0] [0 1 0] [0 1 1] [0 1 1] [0 1 1] [1 1 0]
[1 1 0] [1 1 1] [1 0 0] [1 0 1] [1 1 1] [1 0 1] [1 1 0] [1 1 1] [1 1 0]
and
[0 0 1] [0 0 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [0 1 1] [1 1 1]
[1 1 0] [1 1 1] [0 1 1] [0 1 1] [1 0 1] [1 0 1] [1 1 1] [1 1 1]
[1 1 1] [1 1 1] [1 0 1] [1 1 1] [1 1 0] [1 1 1] [1 1 1] [1 1 1].

J. G. Lee, Almost Distributive Lattice Varieties, Algebra Universalis, 21 (1985), 280-304.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1976.

F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning, B 5 (1978), 31-43. See Table 2.
F. Harary, L. March and R. W. Robinson, On enumerating certain design problems in terms of bicolored graphs with no isolates, Environment and Planning B: Urban Analytics and City Science, 5 (1978), 31-43. [Annotated scanned copy] See Table 2.

Columns k=1..6 are A000012, A024206, A055609, A055082, A055083, A055084.
Row sums give A055192.
See A122083 for another version of this triangle.
Cf. A049312, A048194, A028657, A049311.

Showing 1-10 of 15 results. Next

cp's OEIS Frontend

A123458 Duplicate of A048194.

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A049311 Number of (0,1) matrices with n ones and no zero rows or columns, up to row and column permutations.

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

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A048192 Number of connected chordal graphs on n vertices.

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A055080 Triangle T(n,k) read by rows, giving number of k-member minimal covers of an unlabeled n-set, k=1..n.

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A049312 Number of graphs with a distinguished bipartite block, by number of vertices.

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A056885 Triangle T(n,k) = number of minimal covers of an unlabeled n-set that cover k points of that set uniquely, k=0..n.

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A007776 Number of connected posets with n elements of height 1.

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