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.

Showing 1-10 of 16 results. Next

A317757 Number of non-isomorphic multiset partitions of size n such that the blocks have empty intersection.

Original entry on oeis.org

1, 0, 1, 4, 17, 56, 205, 690, 2446, 8506, 30429, 109449, 402486, 1501424, 5714194, 22132604, 87383864, 351373406, 1439320606, 6003166059, 25488902820, 110125079184, 483987225922, 2162799298162, 9823464989574, 45332196378784, 212459227340403, 1010898241558627, 4881398739414159
Offset: 0

Views

Author

Gus Wiseman, Aug 06 2018

Keywords

Examples

			Non-isomorphic representatives of the a(4) = 17 multiset partitions:
  {1}{234},{2}{111},{2}{113},{11}{22},{11}{23},{12}{34},
  {1}{1}{22},{1}{1}{23},{1}{2}{11},{1}{2}{12},{1}{2}{13},{1}{2}{34},{2}{3}{11},
  {1}{1}{1}{2},{1}{1}{2}{2},{1}{1}{2}{3},{1}{2}{3}{4}.

Links

Crossrefs

Cf. A007716, A035310, A255906, A281116, A317073, A317533.
Cf. A317748, A317751, A317752, A317755.
Cf. A319077, A319748, A319755, A319778, A319781, A319790.

Programs

  • Mathematica
    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];
sysnorm[m_]:=If[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[Length[Union[sysnorm/@Join@@Table[Select[mps[m],Intersection@@#=={}&],{m,strnorm[n]}]]],{n,6}]
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, 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)={EulerT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={my(s=0); forpart(q=n, my(f=prod(i=1, #q, 1 - x^q[i]), u=R(q,n)); s+=permcount(q)*sum(k=0, n, my(c=polcoef(f,k)); if(c, c*polcoef(exp(sum(t=1, n\(k+1), x^(t*k)*u[t], O(x*x^n) ))/if(k,1-x^k,1), n))) ); s/n!} \\ Andrew Howroyd, May 30 2023

Extensions

a(8)-a(10) from Gus Wiseman, Sep 27 2018
a(0)=1 prepended and terms a(11) and beyond from Andrew Howroyd, May 30 2023

A319759 Number of non-isomorphic intersecting multiset partitions of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 2, 13, 49, 199
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(6) = 1 through a(8) = 13 multiset partitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,2},{1,3},{2,3,3}}
   {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3},{2,2,3,3}}
   {{1,2},{1,3},{2,3,3,3}}
   {{1,2},{1,3},{2,3,4,4}}
   {{1,2},{1,3,3},{2,3,3}}
   {{1,2},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4,4}}
   {{1,3},{1,1,2},{2,3,3}}
   {{1,3},{1,2,2},{2,3,3}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,3},{1,2,4},{3,4,4}}
   {{2,4},{1,2,3},{3,4,4}}
   {{2,4},{1,2,5},{3,4,5}}
   {{1,2},{1,3},{2,3},{2,3}}

Crossrefs

Cf. A007716, A283877, A305854, A306006, A317757, A318715, A318717.
Cf. A319752, A319762, A319763, A319764, A319765, A319779, A319781.

A319779 Number of intersecting multiset partitions of weight n whose dual is not an intersecting multiset partition.

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 20, 66, 226, 696, 2156
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

Examples

			Non-isomorphic representatives of the a(4) = 1 through a(6) = 20 multiset partitions:
4: {{1,3},{2,3}}
5: {{1,2},{2,3,3}}
   {{1,3},{2,3,3}}
   {{1,4},{2,3,4}}
   {{3},{1,3},{2,3}}
6: {{1,2},{2,3,3,3}}
   {{1,3},{2,2,3,3}}
   {{1,3},{2,3,3,3}}
   {{1,3},{2,3,4,4}}
   {{1,4},{2,3,4,4}}
   {{1,5},{2,3,4,5}}
   {{1,1,2},{2,3,3}}
   {{1,2,2},{2,3,3}}
   {{1,2,3},{3,4,4}}
   {{1,2,4},{3,4,4}}
   {{1,2,5},{3,4,5}}
   {{1,3,3},{2,3,3}}
   {{1,3,4},{2,3,4}}
   {{2},{1,2},{2,3,3}}
   {{3},{1,3},{2,3,3}}
   {{4},{1,4},{2,3,4}}
   {{1,3},{2,3},{2,3}}
   {{1,3},{2,3},{3,3}}
   {{1,4},{2,4},{3,4}}
   {{3},{3},{1,3},{2,3}}

Crossrefs

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.
Cf. A319775, A319778, A319781, A319783.

A319778 Number of non-isomorphic set systems of weight n with empty intersection whose dual is also a set system with empty intersection.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 13, 28, 72, 181, 483
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The dual of a multiset partition has empty intersection iff no part contains all the vertices.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(6) = 13 multiset partitions:
2: {{1},{2}}
3: {{1},{2},{3}}
4: {{1},{3},{2,3}}
   {{1},{2},{3},{4}}
5: {{1},{2,4},{3,4}}
   {{2},{1,3},{2,3}}
   {{1},{2},{3},{2,3}}
   {{1},{2},{4},{3,4}}
   {{1},{2},{3},{4},{5}}
6: {{3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3}}
   {{1,3},{2,4},{3,4}}
   {{1},{2},{1,3},{2,3}}
   {{1},{2},{3,5},{4,5}}
   {{1},{3},{4},{2,3,4}}
   {{1},{3},{2,4},{3,4}}
   {{1},{4},{2,4},{3,4}}
   {{2},{3},{1,3},{2,3}}
   {{2},{4},{1,2},{3,4}}
   {{1},{2},{3},{4},{3,4}}
   {{1},{2},{3},{5},{4,5}}
   {{1},{2},{3},{4},{5},{6}}

Crossrefs

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757.
Cf. A319775, A319779, A319781, A319783.

A319077 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 1, 3, 12, 37, 130, 428, 1481, 5091, 17979, 64176, 234311, 869645, 3295100, 12720494, 50083996, 200964437, 821845766, 3423694821, 14524845181, 62725701708, 275629610199, 1231863834775, 5597240308384, 25844969339979, 121224757935416, 577359833539428, 2791096628891679
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

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

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(4) = 12 strict multiset partitions with empty intersection:
2: {{1},{2}}
3: {{1},{2,2}}
   {{1},{2,3}}
   {{1},{2},{3}}
4: {{1},{2,2,2}}
   {{1},{2,3,3}}
   {{1},{2,3,4}}
   {{1,1},{2,2}}
   {{1,2},{3,3}}
   {{1,2},{3,4}}
   {{1},{2},{1,2}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{2},{3,4}}
   {{1},{3},{2,3}}
   {{1},{2},{3},{4}}

Links

Crossrefs

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A318715.
Cf. A319748, A319755, A319778, A319781, A319790.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, 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)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
R(q, n)={vector(n, t, subst(x*Ser(K(q, t, n\t)/t), x, x^t))}
a(n)={my(s=0); forpart(q=n, my(f=prod(i=1, #q, 1 - x^q[i]), u=R(q,n)); s+=permcount(q)*sum(k=0, n, my(c=polcoef(f,k)); if(c, c*polcoef(exp(sum(t=1, n\(k+1), x^(t*k)*u[t] - subst(x^(t*k)*u[t] + O(x*x^(n\2)), x, x^2), O(x*x^n) ))*if(k,1+x^k,1), n))) ); s/n!} \\ Andrew Howroyd, May 30 2023

Extensions

Terms a(11) and beyond from Andrew Howroyd, May 30 2023

A319748 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 1, 3, 10, 25, 72, 182, 502, 1332, 3720, 10380, 30142, 88842, 270569, 842957, 2703060, 8885029, 29990388, 103743388, 367811233, 1334925589, 4957151327, 18817501736, 72972267232, 288863499000, 1166486601571, 4802115258807, 20141268290050, 86017885573548, 373852868791639
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

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

Examples

			Non-isomorphic representatives of the a(2) = 1 through a(4) = 10 set multipartitions:
  {{1},{2}}   {{1},{2,3}}     {{1},{2,3,4}}
             {{1},{2},{2}}    {{1,2},{3,4}}
             {{1},{2},{3}}   {{1},{1},{2,3}}
                             {{1},{2},{1,2}}
                             {{1},{2},{3,4}}
                             {{1},{3},{2,3}}
                            {{1},{1},{2},{2}}
                            {{1},{2},{2},{2}}
                            {{1},{2},{3},{3}}
                            {{1},{2},{3},{4}}

Links

Crossrefs

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A319616.
Cf. A319077, A319751, A319755, A319778, A319781, A319791.

Programs

  • PARI
    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)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=R(q,n)); s+=permcount(q)*polcoef(exp(sum(t=1, n, u[t], O(x*x^n))) - exp(sum(t=1, n\2, x^t*u[t], O(x*x^n)))/(1-x), n)); s/n!)} \\ Andrew Howroyd, May 30 2023

Extensions

Terms a(11) and beyond from Andrew Howroyd, May 30 2023

A319790 Number of non-isomorphic connected multiset partitions of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 32, 134, 588, 2335, 9335, 36506, 144263, 571238, 2291894, 9300462, 38303796, 160062325, 679333926, 2927951665, 12817221628, 56974693933, 257132512297, 1177882648846, 5475237760563, 25818721638720, 123473772356785, 598687942799298, 2942344764127039
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

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

Examples

			Non-isomorphic representatives of the a(4) = 1 through a(5) = 5 connected multiset partitions:
4:  {{1},{2},{1,2}}
5: {{1},{2},{1,2,2}}
   {{1},{1,2},{2,2}}
   {{2},{3},{1,2,3}}
   {{2},{1,3},{2,3}}
  {{1},{2},{2},{1,2}}

Links

Crossrefs

Cf. A007716, A007718, A049311, A056156, A283877, A317752, A317755, A317757.
Cf. A319077, A319748, A319755, A319778, A319781, A319791.

Formula

a(n) = A007718(n) - A007716(n) + A317757(n). - Andrew Howroyd, May 31 2023

Extensions

Terms a(11) and beyond from Andrew Howroyd, May 31 2023

A319791 Number of non-isomorphic connected set multipartitions (multisets of sets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 1, 3, 14, 38, 125, 360, 1107, 3297, 10292, 32134, 103759, 340566, 1148150, 3951339, 13925330, 50122316, 184365292, 692145409, 2651444318, 10356184440, 41224744182, 167150406897, 689998967755, 2898493498253, 12384852601731, 53804601888559, 237566072006014
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

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

Examples

			Non-isomorphic representatives of the a(4) = 1 through a(6) = 14 set multipartitions:
4:    {{1},{2},{1,2}}
5:   {{2},{3},{1,2,3}}
     {{2},{1,3},{2,3}}
    {{1},{2},{2},{1,2}}
6:  {{1},{1,4},{2,3,4}}
    {{1},{2,3},{1,2,3}}
    {{3},{4},{1,2,3,4}}
    {{3},{1,4},{2,3,4}}
    {{1,2},{1,3},{2,3}}
    {{1,3},{2,4},{3,4}}
   {{1},{2},{3},{1,2,3}}
   {{1},{2},{1,2},{1,2}}
   {{1},{2},{1,3},{2,3}}
   {{2},{2},{1,3},{2,3}}
   {{2},{3},{3},{1,2,3}}
   {{2},{3},{1,3},{2,3}}
  {{1},{1},{2},{2},{1,2}}
  {{1},{2},{2},{2},{1,2}}

Links

Crossrefs

Cf. A007716, A007718, A049311, A056156, A281116, A283877, A317752, A317755, A317757.
Cf. A319077, A319748, A319755, A319778, A319781, A319790.

Formula

a(n) = A056156(n) - A049311(n) + A319748(n). - Andrew Howroyd, May 31 2023

Extensions

Terms a(11) and beyond from Andrew Howroyd, May 31 2023

A319762 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 4, 9, 24
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

A set multipartition is intersecting if no two parts are disjoint. The weight of a set multipartition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(6) = 1 through a(9) = 9 set multipartitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3,4},{2,3,4}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,4},{1,2,5},{3,4,5}}
   {{1,2},{1,3},{2,3},{2,3}}
9: {{1,3},{1,4,5},{2,3,4,5}}
   {{1,5},{1,6},{2,3,4,5,6}}
   {{2,5},{1,2,6},{3,4,5,6}}
   {{1,2,3},{2,4,5},{3,4,5}}
   {{1,3,5},{2,3,6},{4,5,6}}
   {{1,2},{1,3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1,3},{1,4},{1,4},{2,3,4}}
   {{1,3},{1,4},{3,4},{2,3,4}}

Crossrefs

Cf. A007716, A049311, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.
Cf. A319752, A319759, A319763, A319764, A319765, A319779, A319781.

A319763 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 2, 12, 46, 181
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

A multiset partition is intersecting if no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Examples

			Non-isomorphic representatives of the a(6) = 1 through a(8) = 12 multiset partitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,2},{1,3},{2,3,3}}
   {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3},{2,2,3,3}}
   {{1,2},{1,3},{2,3,3,3}}
   {{1,2},{1,3},{2,3,4,4}}
   {{1,2},{1,3,3},{2,3,3}}
   {{1,2},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4,4}}
   {{1,3},{1,1,2},{2,3,3}}
   {{1,3},{1,2,2},{2,3,3}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,3},{1,2,4},{3,4,4}}
   {{2,4},{1,2,3},{3,4,4}}
   {{2,4},{1,2,5},{3,4,5}}

Crossrefs

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.
Cf. A319752, A319759, A319762, A319764, A319765, A319779, A319781.
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