A006125 -id:A006125 - cp's OEIS frontend

A326210 Number of labeled simple graphs with vertices {1..n} containing a nesting pair of edges, where two edges {a,b}, {c,d} are nesting if a < c and b > d or a > c and b < d.

0, 0, 0, 0, 16, 672, 29888, 2071936, 268204288, 68717285888, 35184350796800, 36028796807919616, 73786976292712960000, 302231454903635611721728, 2475880078570760326175178752, 40564819207303340845566684397568, 1329227995784915872903782635437883392

Offset: 0

Gus Wiseman, Jun 19 2019

nonn

Also simple graphs containing a crossing pair of edges, where two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b.

Also simple graphs such that, if the edges are listed in lexicographic order, their maxima (seconds) are not weakly increasing.

			The a(4) = 16 nesting edge-sets:
  {14,23}
  {12,14,23}
  {13,14,23}
  {14,23,24}
  {14,23,34}
  {12,13,14,23}
  {12,14,23,24}
  {12,14,23,34}
  {13,14,23,24}
  {13,14,23,34}
  {14,23,24,34}
  {12,13,14,23,24}
  {12,13,14,23,34}
  {12,14,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}
The a(4) = 16 crossing edge-sets:
  {13,24}
  {12,13,24}
  {13,14,24}
  {13,23,24}
  {13,24,34}
  {12,13,14,24}
  {12,13,23,24}
  {12,13,24,34}
  {13,14,23,24}
  {13,14,24,34}
  {13,23,24,34}
  {12,13,14,23,24}
  {12,13,14,24,34}
  {12,13,23,24,34}
  {13,14,23,24,34}
  {12,13,14,23,24,34}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Non-nesting graphs are A054726.
Nesting digraphs are A326209.
Nesting (or crossing) set partitions are A016098.
MM-numbers of nesting multiset partitions are A326256.
Cf. A001519, A006125, A095661, A324170, A324172, A324326.
Cf. A326211, A326243, A326244, A326248, A326250.

Table[Length[Select[Subsets[Subsets[Range[n],{2}]],!OrderedQ[Last/@#]&]],{n,0,5}]

seq(n)={my(p=1 + 3/2*x - x^2 - x/2*sqrt(1 - 12*x + 4*x^2 + O(x^n))); concat([0], vector(n, k, 2^binomial(k,2)-polcoef(p,k)))} \\ Andrew Howroyd, Aug 26 2019

A006125(n) = a(n) + A054726(n).

Terms a(7) and beyond from Andrew Howroyd, Aug 26 2019

A339655 Number of non-loop-graphical integer partitions of 2n.

Original entry on oeis.org

0, 0, 1, 3, 7, 14, 28, 51, 91, 156, 260, 425, 680, 1068, 1654, 2524, 3802, 5668, 8350, 12190, 17634, 25306, 36011, 50902, 71441, 99642

Offset: 0

Gus Wiseman, Dec 14 2020

An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with equal source and target. See A339657 for the Heinz numbers, and A339656 for the complement.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The a(2) = 1 through a(5) = 14 partitions (A = 10):
  (4)  (6)    (8)      (A)
       (4,2)  (4,4)    (5,5)
       (5,1)  (5,3)    (6,4)
              (6,2)    (7,3)
              (7,1)    (8,2)
              (5,2,1)  (9,1)
              (6,1,1)  (5,3,2)
                       (5,4,1)
                       (6,2,2)
                       (6,3,1)
                       (7,2,1)
                       (8,1,1)
                       (6,2,1,1)
                       (7,1,1,1)
For example, the seven normal loop-multigraphs with degrees y = (5,3,2) are:
  {{1,1},{1,1},{1,2},{2,2},{3,3}}
  {{1,1},{1,1},{1,2},{2,3},{2,3}}
  {{1,1},{1,1},{1,3},{2,2},{2,3}}
  {{1,1},{1,2},{1,2},{1,2},{3,3}}
  {{1,1},{1,2},{1,2},{1,3},{2,3}}
  {{1,1},{1,2},{1,3},{1,3},{2,2}}
  {{1,2},{1,2},{1,2},{1,3},{1,3}},
but since none of these is a loop-graph (because they are not strict), y is counted under a(5).

Eric Weisstein's World of Mathematics, Graphical partition.

A001358 lists semiprimes, with squarefree case A006881.
A006125 counts labeled graphs, with covering case A006129.
A062740 counts labeled connected loop-graphs.
A101048 counts partitions into semiprimes.
A320461 ranks normal loop-graphs.
A322661 counts covering loop-graphs.
A320655 counts factorizations into semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A339655 (this sequence) counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A007717, A025065, A147878, A320732, A320921, A338898, A338902, A338912, A338913, A339659, A339660, A339662.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[mpsbin[#],UnsameQ@@#&]=={}&]],{n,0,5}]

A058696(n) = a(n) + A339656(n).

a(7)-a(25) from Andrew Howroyd, Jan 10 2024

A339656 Number of loop-graphical integer partitions of 2n.

Original entry on oeis.org

1, 2, 4, 8, 15, 28, 49, 84, 140, 229, 367, 577, 895, 1368, 2064, 3080, 4547, 6642, 9627, 13825, 19704, 27868, 39164, 54656, 75832, 104584

Offset: 0

Gus Wiseman, Dec 14 2020

An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. See A339658 for the Heinz numbers, and A339655 for the complement.
The following are equivalent characteristics for any positive integer n:
(1) the multiset of prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
(2) n can be factored into distinct semiprimes;
(3) the unordered prime signature of n is loop-graphical.

			The a(0) = 1 through a(4) = 15 partitions:
  ()  (2)    (2,2)      (3,3)          (3,3,2)
      (1,1)  (3,1)      (2,2,2)        (4,2,2)
             (2,1,1)    (3,2,1)        (4,3,1)
             (1,1,1,1)  (4,1,1)        (2,2,2,2)
                        (2,2,1,1)      (3,2,2,1)
                        (3,1,1,1)      (3,3,1,1)
                        (2,1,1,1,1)    (4,2,1,1)
                        (1,1,1,1,1,1)  (5,1,1,1)
                                       (2,2,2,1,1)
                                       (3,2,1,1,1)
                                       (4,1,1,1,1)
                                       (2,2,1,1,1,1)
                                       (3,1,1,1,1,1)
                                       (2,1,1,1,1,1,1)
                                       (1,1,1,1,1,1,1,1)
For example, there are four possible loop-graphs with degrees y = (2,2,1,1), namely
  {{1,1},{2,2},{3,4}}
  {{1,1},{2,3},{2,4}}
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,4},{2,3}}
  {{1,3},{1,4},{2,2}},
so y is counted under a(3). On the other hand, there are two possible loop-multigraphs with degrees z = (4,2), namely
  {{1,1},{1,1},{2,2}}
  {{1,1},{1,2},{1,2}},
but neither of these is a loop-graph, so z is not counted under a(3).

Eric Weisstein's World of Mathematics, Graphical partition.

A339658 ranks these partitions.
A001358 lists semiprimes, with squarefree case A006881.
A006125 counts labeled graphs, with covering case A006129.
A027187 counts partitions of even length, ranked by A028260.
A062740 counts labeled connected loop-graphs.
A320461 ranks normal loop-graphs.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A322661 counts covering loop-graphs.
A339845 counts the same partitions by length, or A339844 with zeros.
The following count vertex-degree partitions and give their Heinz numbers:
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A000569 counts graphical partitions (A320922).
- A058696 counts partitions of 2n (A300061).
- A209816 counts multigraphical partitions (A320924).
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
- A339617 counts non-graphical partitions of 2n (A339618).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 [this sequence] counts loop-graphical partitions (A339658).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001222, A025065, A095268, A101048, A320656, A320921, A338902, A338912, A338913, A339659.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@spsbin[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[mpsbin[#],UnsameQ@@#&]!={}&]],{n,0,5}]

A058696(n) = a(n) + A339655(n).

a(8)-a(25) from Andrew Howroyd, Jan 10 2024

A098694 Double-superfactorials: a(n) = Product_{k=1..n} (2k)!.

Original entry on oeis.org

1, 2, 48, 34560, 1393459200, 5056584744960000, 2422112183371431936000000, 211155601241022491077587763200000000, 4417964278440225627098723475313498521600000000000

Offset: 0

Ralf Stephan, Sep 22 2004

nonn

Hankel transform of double factorial numbers A001147. - Paul Barry, Jan 28 2008

Hankel transform of A112934(n+1). - Paul Barry, Dec 04 2009

Seiichi Manyama, Table of n, a(n) for n = 0..28
C. Radoux, Déterminants de Hankel et théorème de Sylvester, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.

Cf. A000178, A010050, A074962, A168467, A268504, A268505, A268506, A271946, A271947.

[&*[ Factorial(2*k): k in [0..n] ]: n in [0..10]]; // Vincenzo Librandi, Dec 11 2016

Table[Product[(2k)!,{k,1,n}],{n,0,10}] (* Vaclav Kotesovec, Nov 13 2014 *)

a(n) = prod(k=1, n, (2*k)!); \\ Michel Marcus, Dec 11 2016

from math import prod
def A098694(n): return prod(((k+1)*((k<<1)+1)<<1)**(n-k) for k in range(1,n+1))<Chai Wah Wu, Nov 26 2023

a(n) = Product_{k=0..n} (2*(k+1)*(2*k+1))^(n-k). - Paul Barry, Jan 28 2008
a(n) = A000178(n)*A057863(n)*A006125(n+1) = A121835(n)*A006125(n+1). - Paul Barry, Jan 28 2008
G.f.: G(0)/(2*x)-1/x, where G(k)= 1  + 1/(1 - 1/(1 + 1/(2*k+2)!/x/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 14 2013
a(n) ~ 2^(n^2+2*n+17/24) * n^(n^2+3*n/2+11/24) * Pi^((n+1)/2) / (A^(1/2) * exp(3*n^2/2+3*n/2-1/24)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - Vaclav Kotesovec, Nov 13 2014
a(n) = A^(3/2)*2^(n^2+n-1/24)*Pi^(-n/2-1/4)*G(n+3/2)*G(n+2)/exp(1/8), where G(n) is the Barnes G-function and A is the Glaisher-Kinkelin constant. - Ilya Gutkovskiy, Dec 11 2016
a(n) = A000178(2*n + 1) / A168467(n). - Vaclav Kotesovec, Oct 28 2017
For n > 0, a(n) = 2^((n+1)/2) * n * sqrt(BarnesG(2*n)*Gamma(n)) * Gamma(2*n). - Vaclav Kotesovec, Nov 27 2024

A136653 G.f.: A(x) satisfies: coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2).

Original entry on oeis.org

1, 1, 1, 4, 39, 748, 27162, 1880872, 252273611, 66358216668, 34506398937158, 35644762692112792, 73356520492898454022, 301274559225693420690360, 2471654510727312089903896948, 40527708183358718551543295827536, 1328579216048284168977214446788083699

Offset: 0

Paul D. Hanna, Jan 15 2008

nonn

a(n) is the number of graphs on vertices 1,...,n such that, when these vertices are arranged counterclockwise around a circle and edges are drawn as straight line segments, the resulting diagram is connected. - Jonathan Novak (j2novak(AT)math.uwaterloo.ca), Apr 30 2010

In this interpretation, both intersecting (set theoretically) and crossing (topologically) edges are considered connected. - Gus Wiseman, Feb 23 2019

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 39*x^4 + 748*x^5 + 27162*x^6 +...
Let F(x) = 1 + x + 2*x^2 + 8*x^3 + 64*x^4 + 1024*x^5 +...+ 2^(n*(n-1)/2)*x^n +..
then A(x) = F(x/A(x)), A(x*F(x)) = F(x).
Coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2),
as can be seen by the main diagonal in the array of
coefficients in the initial powers of A(x):
A^1: [(1), 1, 1, 4, 39, 748, 27162, 1880872, 252273611,...;
A^2: [1, (2), 3, 10, 87, 1582, 55914, 3817876, 508370795,...;
A^3: [1, 3, (6), 19, 147, 2517, 86398, 5813550, 768378627,...;
A^4: [1, 4, 10, (32), 223, 3572, 118778, 7870640, 1032387787,...;
A^5: [1, 5, 15, 50, (320), 4771, 153245, 9992130, 1300492845,...;
A^6: [1, 6, 21, 74, 444, (6144), 190023, 12181278, 1572792585,...;
A^7: [1, 7, 28, 105, 602, 7728, (229376), 14441659, 1849390375,...;
A^8: [1, 8, 36, 144, 802, 9568, 271616, (16777216), 2130394591,...;
A^9: [1, 9, 45, 192, 1053, 11718, 317112, 19192320, (2415919104),...;
dividing each diagonal term in row n by (n+1) gives 2^(n*(n-1)/2).
The diagonal above the main diagonal gives coefficients of l.g.f.:
log(F(x)) = x + 3*x^2/2 + 19*x^3/3 + 223*x^4/4 + 4771*x^5/5 +...

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 39 graphs connected by overlaps or crossings.

Cf. A136652 (log(A(x))), A136654.
Cf. A000699, A002662, A006125, A007297, A016098, A054726, A099947, A306438.
Cf. A324166, A324169, A324172, A324173, A324327, A324328.

max = 15; s = x*Sum[2^(k*(k-1)/2)*x^k, {k, 0, max}] + O[x]^(max+2); x/InverseSeries[s] + O[x]^(max+1) // CoefficientList[#, x]& (* Jean-François Alcover, Sep 03 2017 *)
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
bicmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],Intersection@@#!={}&],Select[Subsets[stn,{2}],croXQ]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[bicmpts[#]]<=1]&]],{n,0,5}] (* Gus Wiseman, Feb 23 2019 *)

a(n)=polcoeff(x/serreverse(x*sum(k=0,n,2^(k*(k-1)/2)*x^k +x*O(x^n))),n)

G.f.: A(x) = x/Series_Reversion( x*Sum_{k=0..n} 2^(k(k-1)/2)*x^k ).

Equals the free cumulant sequence corresponding to A006125. - Jonathan Novak (j2novak(AT)math.uwaterloo.ca), Apr 30 2010

Name changed and part of prior name moved to formula section by Paul D. Hanna, Sep 19 2013

A369197 Number of labeled connected loop-graphs with n vertices, none isolated, and at most n edges.

Original entry on oeis.org

1, 1, 3, 13, 95, 972, 12732, 202751, 3795864, 81609030, 1980107840, 53497226337, 1592294308992, 51758060711792, 1824081614046720, 69272000503031475, 2819906639193992192, 122488526636380368714, 5654657850859704139776, 276462849597009068108405, 14270030377126199463936000

Offset: 0

Gus Wiseman, Jan 17 2024

nonn

			The a(0) = 0 through a(3) = 13 loop-graphs (loops shown as singletons):
  .  {{1}}  {{1,2}}      {{1,2},{1,3}}
            {{1},{1,2}}  {{1,2},{2,3}}
            {{2},{1,2}}  {{1,3},{2,3}}
                         {{1},{1,2},{1,3}}
                         {{1},{1,2},{2,3}}
                         {{1},{1,3},{2,3}}
                         {{2},{1,2},{1,3}}
                         {{2},{1,2},{2,3}}
                         {{2},{1,3},{2,3}}
                         {{3},{1,2},{1,3}}
                         {{3},{1,2},{2,3}}
                         {{3},{1,3},{2,3}}
                         {{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..200

The minimal case is A000272.
Connected case of A066383 and A369196, loopless A369192 and A369193.
The loopless case is A129271, connected case of A369191.
The case of equality is A368951, connected case of A368597.
This is the connected case of A369194.
A000085, A100861, A111924 count set partitions into singletons or pairs.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts (simple) graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A054548 counts graphs covering n vertices with k edges, with loops A369199.
A062740 counts connected loop-graphs.
A322661 counts covering loop-graphs, unlabeled A322700.
A368927 counts choosable loop-graphs, covering A369140.
A369141 counts non-choosable loop-graphs, covering A369142.
Cf. A000169, A000666, A001429, A005703, A006649, A057500, A116508, A140638, A143543, A367863, A369145.

seq(n)={my(t=-lambertw(-x + O(x*x^n))); Vec(serlaplace(log(1/(1-t))/2 + 3*t/2 - 3*t^2/4 + 1 - x))} \\ Andrew Howroyd, Feb 02 2024

Logarithmic transform of A368927.
From Andrew Howroyd, Feb 02 2024: (Start)
a(n) = A000169(n) + A129271(n).
E.g.f.: log(1/(1-T(x)))/2 + 3*T(x)/2 - 3*T(x)^2/4 + 1 - x, where T(x) is the e.g.f. of A000169. (End)

a(0) changed to 1 and a(7) onwards from Andrew Howroyd, Feb 02 2024

A144958 Number of unlabeled acyclic graphs covering n vertices.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 10, 17, 39, 77, 176, 381, 891, 2057, 4941, 11915, 29391, 73058, 184236, 468330, 1202349, 3108760, 8097518, 21218776, 55925742, 148146312, 394300662, 1053929982, 2828250002, 7617271738, 20584886435, 55802753243

Offset: 0

Washington Bomfim, Sep 27 2008

nonn

a(n) is the number of forests with n unlabeled nodes without isolated vertices. This follows from the fact that for n>0 A005195(n-1) counts the forests with one or more isolated nodes.

The labeled version is A105784. The connected case is A000055. This is the covering case of A005195. - Gus Wiseman, Apr 29 2024

			From _Gus Wiseman_, Apr 29 2024: (Start)
Edge-sets of non-isomorphic representatives of the a(0) = 1 through a(5) = 4 forests:
  {}    .    {12}    {13,23}    {12,34}       {12,35,45}
                                {13,24,34}    {13,24,35,45}
                                {14,24,34}    {14,25,35,45}
                                              {15,25,35,45}
(End)

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The connected case is A000055.
This is the covering case of A005195, labeled A001858.
The labeled version is A105784.
For triangles instead of cycles we have A372169, non-covering A006785.
Unique cycle: A372191 (lab A372195), non-covering A236570 (lab A372193).
A006125 counts simple graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
Cf. A000272, A053530, A054548, A137917, A144959, A372174.

brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}]]];
cyc[y_]:=Select[Join@@Table[Select[Join@@Permutations/@Subsets[Union@@y,{k}],And@@Table[MemberQ[Sort/@y,Sort[{#[[i]],#[[If[i==k,1,i+1]]]}]],{i,k}]&],{k,3,Length[y]}],Min@@#==First[#]&];
Table[Length[Union[Union[brute/@Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&&Length[cyc[#]]==0&]]]],{n,0,5}] (* Gus Wiseman, Apr 29 2024 *)

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}
seq(n)={my(t=TreeGf(n), v=EulerT(Vec(t - t^2/2 + subst(t,x,x^2)/2))); concat([1,0], vector(#v-1, i, v[i+1]-v[i]))} \\ Andrew Howroyd, Aug 01 2024

a(n) = A005195(n) - A005195(n-1).

Name changed and 1 prepended by Gus Wiseman, Apr 29 2024.

A309356 MM-numbers of labeled simple covering graphs.

Original entry on oeis.org

1, 13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 199, 233, 257, 269, 271, 293, 313, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 559, 577, 607, 611, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 907, 929, 937, 947, 949

Offset: 1

Gus Wiseman, Jul 25 2019

nonn

First differs from A322551 in having 377.
Also products of distinct elements of A322551.
A multiset multisystem is a finite multiset of finite multisets. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.
Covering means there are no isolated vertices, i.e., the vertex set is the union of the edge set.

			The sequence of edge sets together with their MM-numbers begins:
    1: {}
   13: {{1,2}}
   29: {{1,3}}
   43: {{1,4}}
   47: {{2,3}}
   73: {{2,4}}
   79: {{1,5}}
  101: {{1,6}}
  137: {{2,5}}
  139: {{1,7}}
  149: {{3,4}}
  163: {{1,8}}
  167: {{2,6}}
  199: {{1,9}}
  233: {{2,7}}
  257: {{3,5}}
  269: {{2,8}}
  271: {{1,10}}
  293: {{1,11}}
  313: {{3,6}}
  347: {{2,9}}
  373: {{1,12}}
  377: {{1,2},{1,3}}
  389: {{4,5}}
  421: {{1,13}}

Simple graphs are A006125.
The case for BII-numbers is A326788.
Cf. A001222, A001358, A006129, A056239, A112798, A302242, A320458, A322551.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],And[SquareFreeQ[#],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#])]&]

A326244 Number of labeled n-vertex simple graphs without crossing or nesting edges.

Original entry on oeis.org

1, 1, 2, 8, 36, 160, 704, 3088, 13536, 59328, 260032, 1139712

Offset: 0

Gus Wiseman, Jun 20 2019

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d.

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.
Gus Wiseman, The a(4) = 36 non-crossing, non-nesting simple labeled graphs.

The case for set partitions is A001519.
Simple graphs with crossing or nesting edges are A326279.
Cf. A000088, A000108, A006125, A016098, A054726, A095661, A117662.
Cf. A326209, A326210, A326250, A326255, A326256.

croXQ[stn_]:=MatchQ[stn,{_,{x_,y_},_,{z_,t_},_}/;x_,{x_,y_},_,{z_,t_},_}/;x

A006125(n) = a(n) + A326279(n).
Conjectures from Colin Barker, Jun 28 2019: (Start)
G.f.: (1 - x)*(1 - 4*x) / (1 - 6*x + 8*x^2 - 4*x^3).
a(n) = 6*a(n-1) - 8*a(n-2) + 4*a(n-3) for n>2.
(End)

A327125 Triangle read by rows where T(n,k) is the number of labeled simple graphs with n vertices and cut-connectivity k.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 4, 3, 0, 1, 26, 28, 9, 0, 1, 296, 490, 212, 25, 0, 1, 6064, 15336, 9600, 1692, 75, 0, 1, 230896

Offset: 0

Gus Wiseman, Aug 25 2019

We define the cut-connectivity of a graph to be the minimum number of vertices that must be removed (along with any incident edges) to obtain a disconnected or empty graph, with the exception that a graph with one vertex and no edges has cut-connectivity 1. Except for complete graphs, this is the same as vertex-connectivity.

			Triangle begins:
    1
    0   1
    1   0   1
    4   3   0   1
   26  28   9   0   1
  296 490 212  25   0   1

After the first column, same as A327126.
The unlabeled version is A327127.
Row sums are A006125.
Column k = 0 is A054592, if we assume A054592(0) = 1.
Column k = 1 is A327114, if we assume A327114(1) = 1.
Row sums without the first column are A001187.
Row sums without the first two columns are A013922.
Different from A327069.
Cf. A259862, A322389, A326786, A327082, A327098, A327100, A327101.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
cutConnSys[vts_,eds_]:=If[Length[vts]==1,1,Min@@Length/@Select[Subsets[vts],Function[del,csm[DeleteCases[DeleteCases[eds,Alternatives@@del,{2}],{}]]!={Complement[vts,del]}]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],cutConnSys[Range[n],#]==k&]],{n,0,4},{k,0,n}]

a(21)-a(28) from Robert Price, May 20 2021

a(1) and a(2) corrected by Robert Price, May 20 2021

cp's OEIS Frontend

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A136653 G.f.: A(x) satisfies: coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2).

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