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 23 results. Next

A306186 Array read by antidiagonals upwards where A(n, k) is the number of non-isomorphic multiset partitions of weight n with k levels of brackets.

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 5, 10, 6, 1, 7, 33, 21, 8, 1, 11, 91, 104, 36, 10, 1, 15, 298, 452, 238, 55, 12, 1, 22, 910, 2335, 1430, 455, 78, 14, 1, 30, 3017, 11992, 10179, 3505, 775, 105, 16, 1, 42, 9945, 66810, 74299, 31881, 7297, 1218, 136, 18, 1, 56
Offset: 1

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Examples

			Array begins:
      k=1:  k=2:  k=3:  k=4:  k=5:  k=6:
  n=1:  1     1     1     1     1     1
  n=2:  2     4     6     8    10    12
  n=3:  3    10    21    36    55    78
  n=4:  5    33   104   238   455   775
  n=5:  7    91   452  1430  3505  7297
  n=6: 11   298  2335 10179 31881 80897
Non-isomorphic representatives of the A(3,3) = 21 multiset partitions:
  {{111}}          {{112}}          {{123}}
  {{1}{11}}        {{1}{12}}        {{1}{23}}
  {{1}}{{11}}      {{2}{11}}        {{1}}{{23}}
  {{1}{1}{1}}      {{1}}{{12}}      {{1}{2}{3}}
  {{1}}{{1}{1}}    {{1}{1}{2}}      {{1}}{{2}{3}}
  {{1}}{{1}}{{1}}  {{2}}{{11}}      {{1}}{{2}}{{3}}
                   {{1}}{{1}{2}}
                   {{2}}{{1}{1}}
                   {{1}}{{1}}{{2}}

Crossrefs

Columns: A000041 (k=1), A007716 (k=2), A318566 (k=3).
Rows: A000012 (n=1), A005843 (n=2), A014105 (n=3).
Cf. A002846, A096751, A144150, A290353, A317533, A317791, A323718, A323719.

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]]]];
undats[m_]:=Union[DeleteCases[Cases[m,_?AtomQ,{0,Infinity},Heads->True],List]];
expnorm[m_]:=If[Length[undats[m]]==0,m,If[undats[m]!=Range[Max@@undats[m]],expnorm[m/.Apply[Rule,Table[{undats[m][[i]],i},{i,Length[undats[m]]}],{1}]],First[Sort[expnorm[m,1]]]]];
expnorm[m_,aft_]:=If[Length[undats[m]]<=aft,{m},With[{mx=Table[Count[m,i,{0,Infinity},Heads->True],{i,Select[undats[m],#1>=aft&]}]},Union@@(expnorm[#1,aft+1]&)/@Union[Table[MapAt[Sort,m/.{par+aft-1->aft,aft->par+aft-1},Position[m,[__]]],{par,First/@Position[mx,Max[mx]]}]]]];
strnorm[n_]:=(Flatten[MapIndexed[Table[#2,{#1}]&,#1]]&)/@IntegerPartitions[n];
kmp[n_,k_]:=kmp[n,k]=If[k==1,strnorm[n],Union[expnorm/@Join@@mps/@kmp[n,k-1]]];
Table[Length[kmp[sum-k,k]],{sum,1,7},{k,1,sum-1}]

Extensions

a(46)-a(56) from Robert Price, May 11 2021

A318565 Number of multiset partitions of multiset partitions of strongly normal multisets of size n.

Original entry on oeis.org

1, 6, 27, 169, 1029, 7817, 61006, 547537, 5202009, 54506262, 606311524, 7299051826, 92985064466, 1264720212352, 18137495642192, 275078184766323, 4379514178076452, 73235806332442156, 1280229713195027792, 23381809052104639236, 444740694108284116235, 8801030741502964613534
Offset: 1

Views

Author

Gus Wiseman, Aug 29 2018

Keywords

Comments

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities.

Examples

			The a(2) = 6 multiset partitions of multiset partitions:
  {{{1,1}}}
  {{{1,2}}}
  {{{1},{1}}}
  {{{1},{2}}}
  {{{1}},{{1}}}
  {{{1}},{{2}}}

Crossrefs

Cf. A001970, A007716, A050336, A255906, A269134, A317533, A317791, A318564, A318566.

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];
Table[Sum[Length[mps[m]],{m,Join@@mps/@strnorm[n]}],{n,6}]
  • PARI
    \\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); StronglyNormalLabelingsSeq(sExp(sExp(A))-1)} \\ Andrew Howroyd, Dec 30 2020

Extensions

Terms a(9) and beyond from Andrew Howroyd, Dec 30 2020

A323787 Number of non-isomorphic multiset partitions of strict multiset partitions of weight n.

Original entry on oeis.org

1, 1, 4, 14, 56, 219, 1001, 4588
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

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(1) = 1 through a(3) = 14 multiset partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{2}}    {{123}}
         {{1}}{{2}}  {{1}{11}}
                     {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}

Crossrefs

Cf. A002846, A005121, A007716, A050343, A213427, A269134, A283877, A306186, A316980, A317791, A318564, A318565, A318566, A318812.
Cf. A323788, A323789, A323790, A323791, A323792, A323793, A323794, A323795.

A323795 Number of non-isomorphic weight-n sets of non-overlapping sets of sets.

Original entry on oeis.org

1, 1, 3, 8, 27, 82, 310, 1163
Offset: 0

Views

Author

Gus Wiseman, Jan 28 2019

Keywords

Comments

Also the number of non-isomorphic set partitions of set-systems of weight n.
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(1) = 1 through a(4) = 27 multiset partitions:
  {{1}}  {{12}}      {{123}}          {{1234}}
         {{1}{2}}    {{1}{12}}        {{1}{123}}
         {{1}}{{2}}  {{1}{23}}        {{12}{13}}
                     {{1}}{{12}}      {{1}{234}}
                     {{1}}{{23}}      {{12}{34}}
                     {{1}{2}{3}}      {{1}}{{123}}
                     {{1}}{{2}{3}}    {{1}{2}{12}}
                     {{1}}{{2}}{{3}}  {{1}{2}{13}}
                                      {{12}}{{13}}
                                      {{1}}{{234}}
                                      {{1}{2}{34}}
                                      {{12}}{{34}}
                                      {{1}}{{2}{12}}
                                      {{12}}{{1}{2}}
                                      {{1}}{{2}{13}}
                                      {{12}}{{1}{3}}
                                      {{1}}{{2}{34}}
                                      {{1}{2}{3}{4}}
                                      {{12}}{{3}{4}}
                                      {{2}}{{1}{13}}
                                      {{1}}{{2}}{{12}}
                                      {{1}}{{2}}{{13}}
                                      {{1}}{{2}}{{34}}
                                      {{1}}{{2}{3}{4}}
                                      {{1}{2}}{{3}{4}}
                                      {{1}}{{2}}{{3}{4}}
                                      {{1}}{{2}}{{3}}{{4}}

Crossrefs

Cf. A004111, A007716, A049311, A050326, A050343, A283877, A306186, A318566.
Cf. A323787, A323788, A323789, A323790, A323791, A323792, A323793, A323794.

A318564 Number of multiset partitions of multiset partitions of normal multisets of size n.

Original entry on oeis.org

1, 6, 36, 274, 2408, 24440, 279172, 3542798, 49354816, 747851112, 12231881948, 214593346534, 4016624367288, 79843503990710, 1678916979373760, 37215518578700028, 866953456654946948, 21167221410812128266, 540346299720320080828, 14390314687100383124540, 399023209689817997883900
Offset: 1

Views

Author

Gus Wiseman, Aug 29 2018

Keywords

Comments

A multiset is normal if it spans an initial interval of positive integers.

Examples

			The a(2) = 6 multiset partitions of multiset partitions:
  {{{1,1}}}
  {{{1,2}}}
  {{{1},{1}}}
  {{{1},{2}}}
  {{{1}},{{1}}}
  {{{1}},{{2}}}

Crossrefs

Cf. A001970, A007716, A050336, A255906, A269134, A317533, A317791, A318566.

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]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Sum[Length[mps[m]],{m,Join@@mps/@allnorm[n]}],{n,6}]
  • PARI
    \\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=symGroupSeries(n)); NormalLabelingsSeq(sExp(sExp(A))-1)} \\ Andrew Howroyd, Jan 01 2021

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jan 01 2021

A323790 Number of non-isomorphic weight-n sets of sets of sets.

Original entry on oeis.org

1, 1, 3, 9, 33, 113, 474, 1985
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

Non-isomorphic sets of sets are counted by A283877.
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(1) = 1 through a(3) = 9 sets of sets of sets:
  {{1}}  {{12}}      {{123}}
         {{1}{2}}    {{1}{12}}
         {{1}}{{2}}  {{1}{23}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}
Non-isomorphic representatives of the a(4) = 33 sets of sets of sets:
  {{1234}}             {{1}{123}}         {{1}{2}{12}}       {{1}}{{1}{12}}
  {{1}{234}}           {{12}{13}}         {{1}}{{2}{12}}
  {{12}{34}}           {{1}}{{123}}       {{12}}{{1}{2}}
  {{1}}{{234}}         {{1}{2}{13}}       {{1}}{{2}}{{12}}
  {{1}{2}{34}}         {{12}}{{13}}       {{1}}{{2}}{{1}{2}}
  {{12}}{{34}}         {{1}}{{1}{23}}
  {{1}}{{2}{34}}       {{1}}{{2}{13}}
  {{1}{2}{3}{4}}       {{12}}{{1}{3}}
  {{12}}{{3}{4}}       {{2}}{{1}{13}}
  {{1}}{{2}}{{34}}     {{1}}{{1}{2}{3}}
  {{1}}{{2}{3}{4}}     {{1}}{{2}}{{13}}
  {{1}{2}}{{3}{4}}     {{1}{2}}{{1}{3}}
  {{1}}{{2}}{{3}{4}}   {{1}}{{2}}{{1}{3}}
  {{1}}{{2}}{{3}}{{4}}

Crossrefs

Cf. A004111, A007716, A049311, A050326, A050343, A283877, A306186, A316980, A318564, A318565, A318566, A318812.
Cf. A323787, A323788, A323789, A323791, A323792, A323793, A323794, A323795.

A318559 Number of combinatory separations of the multiset of prime factors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 3, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 3, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 8, 1, 2, 4, 11, 2, 3, 1, 4, 2, 3, 1, 15, 1, 2, 4, 4, 2, 3, 1, 12, 5, 2, 1, 8, 2, 2
Offset: 1

Views

Author

Gus Wiseman, Aug 28 2018

Keywords

Comments

A multiset is normal if it spans an initial interval of positive integers. The type of a multiset is the unique normal multiset that has the same sequence of multiplicities when its entries are taken in increasing order. For example the type of 335556 is 112223. A (headless) combinatory separation of a multiset m is a multiset of normal multisets {t_1,...,t_k} such that there exist multisets {s_1,...,s_k} with multiset union m and such that s_i has type t_i for each i = 1...k.

Examples

			The a(60) = 8 combinatory separations of {2,2,3,5}:
  {1123},
  {1,112}, {1,123}, {11,12}, {12,12},
  {1,1,11}, {1,1,12},
  {1,1,1,1}.

Crossrefs

Cf. A007716, A056239, A112798, A255906, A265947, A269134, A317533, A317791.
Cf. A318393, A318396, A318560, A318562, A318565, A318566, A318567.

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]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
Table[Length[Union[Sort/@Map[normize,mps[primeMS[n]],{2}]]],{n,100}]

A323788 Number of non-isomorphic weight-n sets of multisets of multisets.

Original entry on oeis.org

1, 1, 5, 19, 88, 391, 1995, 10281
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

Also the number of non-isomorphic strict multiset partitions of multiset partitions of weight n.
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(1) = 1 through a(3) = 19 multiset partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{1}}    {{123}}
         {{1}{2}}    {{1}{11}}
         {{1}}{{2}}  {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}{1}{1}}
                     {{1}}{{12}}
                     {{1}{1}{2}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{1}{1}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{2}}{{1}{1}}
                     {{1}}{{2}}{{3}}

Crossrefs

Cf. A005121, A007716, A049311, A050343, A283877, A306186, A316980, A317791, A318564, A318565, A318566, A318812.
Cf. A323787, A323789, A323790, A323791, A323792, A323793, A323794, A323795.

A323789 Number of non-isomorphic weight-n sets of sets of multisets.

Original entry on oeis.org

1, 1, 4, 15, 64, 269, 1310, 6460
Offset: 0

Views

Author

Gus Wiseman, Jan 27 2019

Keywords

Comments

Also the number of non-isomorphic strict multiset partitions, with strict parts, of multiset partitions of weight n.
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(1) = 1 through a(3) = 15 multiset partition partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{2}}    {{123}}
         {{1}}{{2}}  {{1}{11}}
                     {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}

Crossrefs

Cf. A007716, A049311, A050343, A283877, A316980, A317791, A318564, A318565, A318566, A318812.
Cf. A323787, A323788, A323790, A323791, A323792, A323793, A323794, A323795.

A323791 Number of non-isomorphic weight-n sets of multisets of sets.

Original entry on oeis.org

1, 1, 4, 13, 52, 196, 877, 3917
Offset: 0

Views

Author

Gus Wiseman, Jan 27 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(1) = 1 through a(3) = 13 sets of multisets of sets:
  {{1}}  {{12}}      {{123}}
         {{1}{1}}    {{1}{12}}
         {{1}{2}}    {{1}{23}}
         {{1}}{{2}}  {{1}{1}{1}}
                     {{1}}{{12}}
                     {{1}{1}{2}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{1}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{2}}{{1}{1}}
                     {{1}}{{2}}{{3}}

Crossrefs

Cf. A007716, A049311, A050326, A050343, A283877, A306186, A316980, A318564, A318565, A318566, A318812.
Cf. A323787, A323788, A323789, A323790, A323792, A323793, A323794, A323795.
Showing 1-10 of 23 results. Next

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