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 1-4 of 4 results.

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

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

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

A319792 Number of non-isomorphic connected set systems of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 1, 2, 9, 22, 69, 190, 567, 1640, 5025, 15404, 49048, 159074, 531165, 1813627, 6352739, 22759620, 83443086, 312612543, 1196356133, 4672620842, 18615188819, 75593464871, 312729620542, 1317267618429, 5646454341658, 24618309943464, 109123789229297
Offset: 0

Views

Author

Gus Wiseman, Sep 27 2018

Keywords

Comments

The weight of a set system 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) = 9 connected set systems:
4:   {{1},{2},{1,2}}
5:  {{2},{3},{1,2,3}}
    {{2},{1,3},{2,3}}
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},{1,3},{2,3}}
  {{2},{3},{1,3},{2,3}}

Links

Crossrefs

Cf. A007716, A007718, A049311, A056156, A281116, A283877, A300913, A305854, A316980, A317752, A317755, A317757.
Cf. A319077, A319748, A319751, A319755, A319778, A319781, A319790.

Formula

a(n) = A300913(n) - A283877(n) + A319751(n). - Andrew Howroyd, May 31 2023

Extensions

Terms a(11) and beyond from Andrew Howroyd, May 31 2023
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.