A317533 -id:A317533 - cp's OEIS frontend

A318285 Number of non-isomorphic multiset partitions of a multiset whose multiplicities are the prime indices of n.

1, 1, 2, 2, 3, 4, 5, 3, 7, 7, 7, 9, 11, 12, 16, 5, 15, 17, 22, 16, 29, 19, 30, 16, 21, 30, 23, 29, 42, 52, 56, 7, 47, 45, 57, 43, 77, 67, 77, 31, 101, 98, 135, 47, 85, 97, 176, 29, 66, 64, 118, 77, 231, 69, 97, 57, 181, 139, 297, 137, 385, 195, 166, 11, 162, 171, 490, 118

Offset: 1

Gus Wiseman, Aug 23 2018

nonn

			Non-isomorphic representatives of the a(12) = 9 multiset partitions of {1,1,2,3}:
  {{1,1,2,3}}
  {{1},{1,2,3}}
  {{2},{1,1,3}}
  {{1,1},{2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{2},{3},{1,1}}
  {{1},{1},{2},{3}}

Andrew Howroyd, Table of n, a(n) for n = 1..130

Cf. A007716, A181821, A255906, A269134, A305936, A317533, A317791.

Cf. A318283, A318284, A318287, A318362, A318371.

\\ See links in A339645 for combinatorial species functions.
sig(n)={my(f=factor(n), sig=vector(primepi(vecmax(f[,1])))); for(i=1, #f~, sig[primepi(f[i,1])]=f[i,2]); sig}
C(sig)={my(n=sum(i=1, #sig, i*sig[i]), A=Vec(symGroupSeries(n)-1), B=O(x*x^n), c=prod(i=1, #sig, if(sig[i], sApplyCI(A[sig[i]], sig[i], A[i], i), 1))); polcoef(OgfSeries(sCartProd(c*x^n + B, sExp(x*Ser(A) + B))), n)}
a(n)={if(n==1, 1, C(sig(n)))} \\ Andrew Howroyd, Jan 17 2023

a(n) = A317791(A181821(n)).

Terms a(31) and beyond from Andrew Howroyd, Jan 17 2023

A330061 MM-number of the VDD-normalization of the multiset of multisets with MM-number n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 7, 8, 9, 6, 3, 12, 13, 14, 15, 16, 3, 18, 19, 12, 21, 6, 7, 24, 9, 26, 27, 28, 13, 30, 3, 32, 15, 6, 35, 36, 37, 38, 39, 24, 3, 42, 13, 12, 45, 14, 13, 48, 49, 18, 15, 52, 53, 54, 15, 56, 57, 26, 3, 60, 37, 6, 63, 64, 39, 30, 3, 12, 35, 70

Offset: 1

Gus Wiseman, Dec 03 2019

nonn

We define the VDD (vertex-degrees decreasing) normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, then selecting only the representatives whose vertex-degrees are weakly decreasing, and finally taking the least of these representatives, where the ordering of multisets is first by length and then lexicographically.
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}

Wikipedia, Idempotence

This sequence is idempotent and its image/fixed points are A330060.
Non-isomorphic multiset partitions are A007716.
MM-weight is A302242.
Cf. A000612, A055621, A056239, A112798, A283877, A316983, A317533, A320456, A330098, A330102, A330103, A330105.
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sysnorm[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],sysnorm[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[sysnorm[m,1]]]];
sysnorm[m_,aft_]:=If[Length[Union@@m]<=aft,{m},With[{mx=Table[Count[m,i,{2}],{i,Select[Union@@m,#>=aft&]}]},Union@@(sysnorm[#,aft+1]&/@Union[Table[Map[Sort,m/.{par+aft-1->aft,aft->par+aft-1},{0,1}],{par,First/@Position[mx,Max[mx]]}]])]];
Table[Map[Times@@Prime/@#&,sysnorm[primeMS/@primeMS[n]],{0,1}],{n,100}]

A330236 MM-numbers of fully chiral multisets of multisets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 53, 54, 56, 57, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Dec 10 2019

nonn

A multiset of multisets is fully chiral every permutation of the vertices gives a different representative.

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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

			The sequence of all fully chiral multisets of multisets together with their MM-numbers begins:
   1:             18: {}{1}{1}      37: {112}          57: {1}{111}
   2: {}          19: {111}         38: {}{111}        59: {7}
   3: {1}         20: {}{}{2}       39: {1}{12}        61: {122}
   4: {}{}        21: {1}{11}       40: {}{}{}{2}      62: {}{5}
   5: {2}         22: {}{3}         41: {6}            63: {1}{1}{11}
   6: {}{1}       23: {22}          42: {}{1}{11}      64: {}{}{}{}{}{}
   7: {11}        24: {}{}{}{1}     44: {}{}{3}        65: {2}{12}
   8: {}{}{}      25: {2}{2}        45: {1}{1}{2}      67: {8}
   9: {1}{1}      27: {1}{1}{1}     46: {}{22}         68: {}{}{4}
  10: {}{2}       28: {}{}{11}      48: {}{}{}{}{1}    69: {1}{22}
  11: {3}         31: {5}           49: {11}{11}       70: {}{2}{11}
  12: {}{}{1}     32: {}{}{}{}{}    50: {}{2}{2}       71: {113}
  14: {}{11}      34: {}{4}         53: {1111}         72: {}{}{}{1}{1}
  16: {}{}{}{}    35: {2}{11}       54: {}{1}{1}{1}    74: {}{112}
  17: {4}         36: {}{}{1}{1}    56: {}{}{}{11}     75: {1}{2}{2}
The complement starts: {13, 15, 26, 29, 30, 33, 43, 47, 51, 52, 55, 58, 60, 66, 73, 79, 85, 86, 93, 94}.

Costrict (or T_0) factorizations are A316978.
BII-numbers of fully chiral set-systems are A330226.
Non-isomorphic fully chiral multiset partitions are A330227.
Full chiral partitions are A330228.
Fully chiral covering set-systems are A330229.
Fully chiral factorizations are A330235.
Cf. A001055, A007716, A083323, A112798, A303975, A317533, A330098, A330223, A330224, A330232.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Select[Range[100],Length[graprms[primeMS/@primeMS[#]]]==Length[Union@@primeMS/@primeMS[#]]!&]

Numbers n such that A330098(n) = A303975(n)!.

A330105 MM-number of the brute-force normalization of the multiset of multisets with MM-number n.

Original entry on oeis.org

1, 2, 3, 4, 3, 6, 7, 8, 9, 6, 3, 12, 13, 14, 15, 16, 3, 18, 19, 12, 21, 6, 7, 24, 9, 26, 27, 28, 13, 30, 3, 32, 15, 6, 69, 36, 37, 38, 39, 24, 3, 42, 13, 12, 45, 14, 13, 48, 49, 18, 15, 52, 53, 54, 15, 56, 57, 26, 3, 60, 37, 6, 63, 64, 39, 30, 3, 12, 69, 138

Offset: 1

Gus Wiseman, Dec 02 2019

nonn

We define the brute-force normalization of a multiset of multisets to be obtained by first normalizing so that the vertices cover an initial interval of positive integers, then applying all permutations to the vertex set, and finally taking the least representative, where the ordering of multisets is first by length and then lexicographically.
For example, 15301 is the MM-number of {{3},{1,2},{1,1,4}}, which has the following normalizations together with their MM-numbers:
  Brute-force:    43287: {{1},{2,3},{2,2,4}}
  Lexicographic:  43143: {{1},{2,4},{2,2,3}}
  VDD:            15515: {{2},{1,3},{1,1,4}}
  MM:             15265: {{2},{1,4},{1,1,3}}
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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

This sequence is idempotent and its image/fixed points are A330104.
Non-isomorphic multiset partitions are A007716.
Cf. A000612, A055621, A283877, A316983, A317533, A330098, A330103.
Other normalizations: A330061 (VDD MM), A330101 (brute-force BII), A330102 (VDD BII), A330105 (brute-force MM).
Other fixed points:
- Brute-force: A330104 (multisets of multisets), A330107 (multiset partitions), A330099 (set-systems).
- Lexicographic: A330120 (multisets of multisets), A330121 (multiset partitions), A330110 (set-systems).
- VDD: A330060 (multisets of multisets), A330097 (multiset partitions), A330100 (set-systems).
- MM: A330108 (multisets of multisets), A330122 (multiset partitions), A330123 (set-systems).
- BII: A330109 (set-systems).

brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];
brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Table[Map[Times@@Prime/@#&,brute[primeMS/@primeMS[n]],{0,1}],{n,100}]

A320808 Regular tetrangle where T(n,k,i) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n, with i columns.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 1, 0, 2, 4, 0, 1, 5, 4, 0, 1, 5, 5, 5, 0, 0, 1, 0, 2, 4, 0, 2, 10, 8, 0, 1, 9, 13, 7, 0, 1, 5, 12, 9, 7, 0, 0, 1, 0, 3, 6, 0, 3, 16, 12, 0, 2, 24, 33, 16, 0, 1, 14, 36, 29, 12, 0, 1, 9, 23, 29

Offset: 1

Gus Wiseman, Nov 09 2018

			Tetrangle begins:
  1  0    0      0        0          0
     0 1  0 1    0 1      0 1        0 1
          0 1 2  0 1 2    0 2 4      0 2 4
                 0 1 2 3  0 1 5 4    0 2 10 8
                          0 1 5 5 5  0 1 9 13 7
                                     0 1 5 12 9 7

Triangle sums are A007716. Triangle of row sums is A320801. Triangle of column sums is A317533. Triangle of last columns (without its leading column 1,0,0,0,...) is A055884.

Cf. A000219, A008284, A316980, A316983, A318805, A319616, A320796, A321194.

A322115 Triangle read by rows where T(n,k) is the number of unlabeled connected multigraphs with loops with n edges and k vertices.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 2, 1, 6, 11, 9, 3, 1, 9, 25, 34, 20, 6, 1, 12, 52, 104, 99, 49, 11, 1, 16, 94, 274, 387, 298, 118, 23, 1, 20, 162, 645, 1295, 1428, 881, 300, 47, 1, 25, 263, 1399, 3809, 5803, 5088, 2643, 765, 106, 1, 30, 407, 2823, 10187, 20645, 24606, 17872, 7878, 1998, 235

Offset: 0

Gus Wiseman, Nov 26 2018

			Triangle begins:
  1
  1   1
  1   2   1
  1   4   4   2
  1   6  11   9   3
  1   9  25  34  20   6
  1  12  52 104  99  49  11

Andrew Howroyd, Table of n, a(n) for n = 0..1274

Row sums are A007719. Diagonal k = n-1 is A000055.

Cf. A000664, A007716, A007718, A191646, A191970, A275421, A317533, A322114.

EulerT(v)={my(p=exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1); Vec(p/x,-#v)}
InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(serchop( sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i), 1))}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
edges(v,x)={sum(i=2, #v, sum(j=1, i-1, my(g=gcd(v[i],v[j])); g*x^(v[i]*v[j]/g))) + sum(i=1, #v, my(t=v[i]); ((t+1)\2)*x^t + if(t%2, 0, x^(t/2)))}
G(n,m)={my(s=0); forpart(p=n, s+=permcount(p)*EulerT(Vec(edges(p,x) + O(x*x^m), -m))); s/n!}
R(n)={Mat(apply(p->Col(p+O(y^n), -n), InvEulerMT(vector(n, k, 1 + y*Ser(G(k,n-1), y)))))}
{ my(T=R(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Nov 30 2018

Terms a(28) and beyond from Andrew Howroyd, Nov 30 2018

A368409 Number of non-isomorphic connected set-systems of weight n contradicting a strict version of the axiom of choice.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 3, 5, 16, 41, 130

Offset: 0

Gus Wiseman, Dec 25 2023

A set-system is a finite set of finite nonempty sets. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

The axiom of choice says that, given any set of nonempty sets Y, it is possible to choose a set containing an element from each. The strict version requires this set to have the same cardinality as Y, meaning no element is chosen more than once.

			Non-isomorphic representatives of the a(4) = 1 through a(8) = 16 set-systems:
  {1}{2}{12}  .  {1}{2}{13}{23}  {1}{3}{23}{123}    {1}{5}{15}{2345}
                 {1}{2}{3}{123}  {1}{4}{14}{234}    {2}{13}{23}{123}
                 {2}{3}{13}{23}  {2}{3}{23}{123}    {3}{13}{23}{123}
                                 {3}{12}{13}{23}    {3}{4}{34}{1234}
                                 {1}{2}{3}{13}{23}  {1}{2}{13}{24}{34}
                                                    {1}{2}{3}{14}{234}
                                                    {1}{2}{3}{23}{123}
                                                    {1}{2}{3}{4}{1234}
                                                    {1}{3}{4}{14}{234}
                                                    {2}{3}{12}{13}{23}
                                                    {2}{3}{13}{24}{34}
                                                    {2}{3}{14}{24}{34}
                                                    {2}{3}{4}{14}{234}
                                                    {2}{4}{13}{24}{34}
                                                    {3}{4}{13}{24}{34}
                                                    {3}{4}{14}{24}{34}

Wikipedia, Axiom of choice.

For unlabeled graphs we have A140636, connected case of A140637.
For labeled graphs: A140638, connected case of A367867 (complement A133686).
This is the connected case of A368094.
The complement is A368410, connected case of A368095.
Allowing repeats: A368411, connected case of A368097, ranks A355529.
Complement with repeats: A368412, connected case of A368098, ranks A368100.
Allowing repeat edges only: connected case of A368421 (complement A368422).
A000110 counts set partitions, non-isomorphic A000041.
A003465 counts covering set-systems, unlabeled A055621.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A134964, A302545, A306005, A317533, A321405, A326031, A367903, A367907, A368187, A368413.

sps[{}]:={{}}; sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpm[n_]:=Join@@Table[Union[Sort[Sort /@ (#/.x_Integer:>s[[x]])]&/@sps[Range[n]]],{s,Flatten[MapIndexed[Table[#2, {#1}]&,#]]&/@IntegerPartitions[n]}];
brute[m_]:=First[Sort[Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]}, {i,Length[p]}])],{p,Permutations[Union@@m]}]]];
csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}],Length[Intersection@@s[[#]]]>0&]}, If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
Table[Length[Union[brute/@Select[mpm[n], UnsameQ@@#&&And@@UnsameQ@@@#&&Length[csm[#]]==1&&Select[Tuples[#], UnsameQ@@#&]=={}&]]],{n,0,6}]

A330055 Number of non-isomorphic set-systems of weight n with no singletons or endpoints.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 3, 5, 16, 24, 90, 179, 567, 1475, 4623, 13650, 44475, 144110, 492017, 1706956, 6124330, 22442687, 84406276, 324298231, 1273955153, 5106977701, 20885538133, 87046940269, 369534837538, 1596793560371, 7019424870960, 31374394197536, 142514998263015

Offset: 0

Gus Wiseman, Nov 30 2019

nonn

A set-system is a finite set of finite nonempty set of positive integers. A singleton is an edge of size 1. An endpoint is a vertex appearing only once (degree 1). The weight of a set-system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(7) = 1 through a(10) = 16 set-systems:
  {12}{13}{123}  {12}{134}{234}    {12}{134}{1234}    {12}{1345}{2345}
                 {12}{34}{1234}    {123}{124}{134}    {123}{124}{1234}
                 {12}{13}{24}{34}  {12}{13}{14}{234}  {123}{145}{2345}
                                   {12}{13}{23}{123}  {12}{345}{12345}
                                   {12}{13}{24}{134}  {12}{13}{124}{134}
                                                      {12}{13}{124}{234}
                                                      {12}{13}{14}{1234}
                                                      {12}{13}{24}{1234}
                                                      {12}{13}{245}{345}
                                                      {12}{13}{45}{2345}
                                                      {12}{34}{123}{124}
                                                      {12}{34}{125}{345}
                                                      {12}{34}{135}{245}
                                                      {13}{24}{123}{124}
                                                      {12}{13}{14}{23}{24}
                                                      {12}{13}{24}{35}{45}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Wikipedia, Degree (graph theory)

The labeled version is A330056.
The "multi" version is A320665.
Non-isomorphic set-systems with no singletons are A306005.
Non-isomorphic set-systems with no endpoints are A330054.
Non-isomorphic set-systems counted by vertices are A000612.
Non-isomorphic set-systems counted by weight are A283877.
Cf. A007716, A055621, A302545, A317533, A317794, A319559, A321405, A330052, A330058.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={my(g=x*Ser(WeighT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k)))); (1-x)*g-subst(g,x,x^2)}
S(q, t, k)={(x-x^2)*sum(j=1, #q, if(t%q[j]==0, q[j])) + O(x*x^k)}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q)*polcoef(exp(sum(t=1, n, subst(K(q, t, n\t)-S(q,t,n\t),x,x^t)/t )), n)); s/n!)} \\ Andrew Howroyd, Jan 27 2024

a(11) onwards from Andrew Howroyd, Jan 27 2024

A321760 Number of non-isomorphic multiset partitions of weight n with no constant parts or vertices that appear in only one part.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 7, 9, 37, 79, 273, 755, 2648, 8432, 29872, 104624, 384759, 1432655, 5502563, 21533141, 86291313, 352654980, 1471073073, 6253397866, 27083003687, 119399628021, 535591458635, 2443030798539, 11326169401988, 53343974825122, 255121588496338

Offset: 0

Gus Wiseman, Nov 29 2018

nonn

Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n in which every row and column has at least two nonzero entries.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(4) = 1 through a(7) = 9 multiset partitions:
  {{1,2},{1,2}}  {{1,2},{1,2,2}}  {{1,1,2},{1,2,2}}    {{1,1,2},{1,2,2,2}}
                                  {{1,2},{1,1,2,2}}    {{1,2},{1,1,2,2,2}}
                                  {{1,2},{1,2,2,2}}    {{1,2},{1,2,2,2,2}}
                                  {{1,2,2},{1,2,2}}    {{1,2,2},{1,1,2,2}}
                                  {{1,2,3},{1,2,3}}    {{1,2,2},{1,2,2,2}}
                                  {{1,2},{1,2},{1,2}}  {{1,2,3},{1,2,3,3}}
                                  {{1,2},{1,3},{2,3}}  {{1,2},{1,2},{1,2,2}}
                                                       {{1,2},{1,3},{2,3,3}}
                                                       {{1,3},{2,3},{1,2,3}}

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

Row sums of A369286.

Cf. A001970, A007716, A050535, A055884, A120733, A317533, A320665, A320798, A320801, A320808, A321407.

Vec(G(20,1)) \\ G defined in A369286. - Andrew Howroyd, Jan 28 2024

a(11) onwards from Andrew Howroyd, Jan 27 2024

A330052 Number of non-isomorphic set-systems of weight n with at least one endpoint.

Original entry on oeis.org

0, 1, 2, 4, 8, 18, 40, 94, 228, 579, 1508, 4092, 11478, 33337, 100016, 309916, 990008, 3257196, 11021851, 38314009, 136657181, 499570867, 1869792499, 7158070137, 28003286261, 111857491266, 455852284867, 1893959499405, 8017007560487, 34552315237016, 151534813272661

Offset: 0

Gus Wiseman, Nov 30 2019

nonn

A set-system is a finite set of finite nonempty sets of positive integers. An endpoint is a vertex appearing only once (degree 1). The weight of a set-system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 18 multiset partitions:
  {1}  {12}    {123}      {1234}        {12345}
       {1}{2}  {1}{12}    {1}{123}      {1}{1234}
               {1}{23}    {12}{13}      {12}{123}
               {1}{2}{3}  {1}{234}      {12}{134}
                          {12}{34}      {1}{2345}
                          {1}{2}{13}    {12}{345}
                          {1}{2}{34}    {1}{12}{13}
                          {1}{2}{3}{4}  {1}{12}{23}
                                        {1}{12}{34}
                                        {1}{2}{123}
                                        {1}{2}{134}
                                        {1}{2}{345}
                                        {1}{23}{45}
                                        {2}{13}{14}
                                        {1}{2}{3}{12}
                                        {1}{2}{3}{14}
                                        {1}{2}{3}{45}
                                        {1}{2}{3}{4}{5}

Andrew Howroyd, Table of n, a(n) for n = 0..50
Wikipedia, Degree (graph theory)

The complement is counted by A330054.
The multiset partition version is A330058.
Non-isomorphic set-systems with at least one singleton are A330053.
Non-isomorphic set-systems counted by vertices are A000612.
Non-isomorphic set-systems counted by weight are A283877.
Cf. A007716, A055621, A306005, A317533, A317794, A319559, A320665, A330055, A330056.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
brute[{}]:={};brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];
Table[Length[Select[Union[brute/@Join@@mps/@strnorm[n]],UnsameQ@@#&&And@@UnsameQ@@@#&&Min@@Length/@Split[Sort[Join@@#]]==1&]],{n,0,5}]

a(n) = A283877(n) - A330054(n). - Andrew Howroyd, Jan 27 2024

a(11) onwards from Andrew Howroyd, Jan 27 2024

cp's OEIS Frontend

A318285 Number of non-isomorphic multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A330061 MM-number of the VDD-normalization of the multiset of multisets with MM-number n.

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A330236 MM-numbers of fully chiral multisets of multisets.

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A330105 MM-number of the brute-force normalization of the multiset of multisets with MM-number n.

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A320808 Regular tetrangle where T(n,k,i) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n, with i columns.

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A322115 Triangle read by rows where T(n,k) is the number of unlabeled connected multigraphs with loops with n edges and k vertices.

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A368409 Number of non-isomorphic connected set-systems of weight n contradicting a strict version of the axiom of choice.

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A330055 Number of non-isomorphic set-systems of weight n with no singletons or endpoints.

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A321760 Number of non-isomorphic multiset partitions of weight n with no constant parts or vertices that appear in only one part.

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