cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A330196 Number of unlabeled set-systems covering n vertices with no endpoints.

Original entry on oeis.org

1, 0, 1, 20, 1754
Offset: 0

Views

Author

Gus Wiseman, Dec 05 2019

Keywords

Comments

A set-system is a finite set of finite nonempty sets. An endpoint is a vertex appearing only once (degree 1).

Examples

			Non-isomorphic representatives of the a(3) = 20 set-systems:
  {12}{13}{23}
  {1}{23}{123}
  {12}{13}{123}
  {1}{2}{13}{23}
  {1}{2}{3}{123}
  {1}{12}{13}{23}
  {1}{2}{13}{123}
  {1}{12}{13}{123}
  {1}{12}{23}{123}
  {12}{13}{23}{123}
  {1}{2}{3}{12}{13}
  {1}{2}{12}{13}{23}
  {1}{2}{3}{12}{123}
  {1}{2}{12}{13}{123}
  {1}{2}{13}{23}{123}
  {1}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}
  {1}{2}{3}{12}{13}{123}
  {1}{2}{12}{13}{23}{123}
  {1}{2}{3}{12}{13}{23}{123}

Crossrefs

First differences of the non-covering version A330124.
The "multi" version is A302545.
Unlabeled set-systems with no endpoints counted by vertices are A317794.
Unlabeled set-systems with no endpoints counted by weight are A330054.
Unlabeled set-systems counted by vertices are A000612.
Unlabeled set-systems counted by weight are A283877.
Cf. A007716, A016031, A306005, A317795, A320665, A330052, A330053, A330055.

A357873 Numbers whose multiset of prime factors has all non-isomorphic multiset partitions.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73
Offset: 1

Views

Author

Gus Wiseman, Oct 18 2022

Keywords

Comments

These are the positions where A317791 matches A001055.

Examples

			The multiset partitions of the prime indices of 12 are: {{1,1,2}}, {{1},{1,2}}, {{1,1},{2}}, {{1},{1},{2}}, all of which are non-isomorphic, so 12 is in the sequence.
The multiset partitions of the prime indices of 30 are: {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}, of which the middle 3 are isomorphic, so 30 is not in the sequence.

Crossrefs

The complement is A357874.
A001055 counts multiset partitions of prime indices, non-isomorphic A317791.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A000612, A007716, A055621, A283877, A302545, A317533, A321194.

Programs

  • Mathematica
    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]}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@brute/@mps[primeMS[#]]&]

A357874 Numbers whose multiset of prime factors has at least two multiset partitions that are isomorphic.

Original entry on oeis.org

30, 36, 42, 60, 66, 70, 78, 84, 90, 100, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 196, 198, 204, 210, 216, 220, 222, 225, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258, 260, 264, 266, 270
Offset: 1

Views

Author

Gus Wiseman, Oct 18 2022

Keywords

Comments

These are the positions where A317791 differs from A001055.

Examples

			The terms together with their prime indices begin:
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
  100: {1,1,3,3}
For example, the multiset partitions of the prime indices of 36 include {{1},{1,2,2}} and {{2},{1,1,2}}, which are isomorphic, so 36 is in the sequence.

Crossrefs

The complement is A357873.
A001055 counts multiset partitions of prime indices, non-isomorphic A317791.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798.
Cf. A000612, A007716, A055621, A283877, A300913, A302545, A317533, A321194.

Programs

  • Mathematica
    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]}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!UnsameQ@@brute/@mps[primeMS[#]]&]

A302546 a(n) = Sum_{k = 1...n} 2^binomial(n, k).

Original entry on oeis.org

0, 2, 6, 18, 98, 2114, 1114242, 68723671298, 1180735735906024030722, 170141183460507917357914971986913657858, 7237005577335553223087828975127304179197147198604070555943173844710572689410
Offset: 0

Views

Author

Gus Wiseman, Jun 20 2018

Keywords

Crossrefs

Cf. A001315, A003465, A034729, A055621, A302545.

Programs

  • Mathematica
    Table[Sum[2^Binomial[n,d],{d,n}],{n,10}]
  • PARI
    a(n) = sum(k=1, n, 2^binomial(n, k)); \\ Michel Marcus, Jun 21 2018

Formula

a(n) = A001315(n) - 2.

A323786 Number of non-isomorphic weight-n multisets of multisets of non-singleton multisets.

Original entry on oeis.org

1, 0, 2, 3, 19, 39, 200, 615, 2849, 11174, 52377, 239269, 1191090, 6041975, 32275288, 177797719, 1017833092, 6014562272, 36717301665, 230947360981, 1495562098099, 9956230757240, 68070158777759, 477439197541792, 3432259679880648, 25267209686664449
Offset: 0

Views

Author

Gus Wiseman, Jan 28 2019

Keywords

Comments

All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

Examples

			Non-isomorphic representatives of the a(4) = 19 multiset partitions:
  {{1111}}      {{1112}}      {{1123}}      {{1234}}
  {{11}{11}}    {{1122}}      {{11}{23}}    {{12}{34}}
  {{11}}{{11}}  {{11}{12}}    {{12}{13}}    {{12}}{{34}}
                {{11}{22}}    {{11}}{{23}}
                {{12}{12}}    {{12}}{{13}}
                {{11}}{{12}}
                {{11}}{{22}}
                {{12}}{{12}}
Non-isomorphic representatives of the a(5) = 39 multiset partitions:
  {{11111}}      {{11112}}      {{11123}}      {{11234}}      {{12345}}
  {{11}{111}}    {{11122}}      {{11223}}      {{11}{234}}    {{12}{345}}
  {{11}}{{111}}  {{11}{112}}    {{11}{123}}    {{12}{134}}    {{12}}{{345}}
                 {{11}{122}}    {{11}{223}}    {{23}{114}}
                 {{12}{111}}    {{12}{113}}    {{11}}{{234}}
                 {{12}{112}}    {{12}{123}}    {{12}}{{134}}
                 {{22}{111}}    {{13}{122}}    {{23}}{{114}}
                 {{11}}{{112}}  {{23}{111}}
                 {{11}}{{122}}  {{11}}{{123}}
                 {{12}}{{111}}  {{11}}{{223}}
                 {{12}}{{112}}  {{12}}{{113}}
                 {{22}}{{111}}  {{12}}{{123}}
                                {{13}}{{122}}
                                {{23}}{{111}}

Crossrefs

Cf. A007716, A302545, A306186, A317791, A318564, A318566.
Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323793, A323794.

Programs

  • PARI
    \\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NumUnlabeledObjsSeq(sCartProd(sExp(A), sExp(sExp(A-x*sv(1)))))} \\ Andrew Howroyd, Jan 17 2023

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jan 17 2023
Previous Showing 41-45 of 45 results.

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