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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A321229 Number of non-isomorphic connected weight-n multiset partitions with multiset density -1.

Original entry on oeis.org

1, 1, 3, 6, 16, 37, 105, 279, 817, 2387, 7269
Offset: 0

Views

Author

Gus Wiseman, Oct 31 2018

Keywords

Comments

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of 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(1) = 1 through a(5) = 37 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}        {{1,1,1,1,1}}
         {{1,2}}    {{1,2,2}}      {{1,1,2,2}}        {{1,1,2,2,2}}
         {{1},{1}}  {{1,2,3}}      {{1,2,2,2}}        {{1,2,2,2,2}}
                    {{1},{1,1}}    {{1,2,3,3}}        {{1,2,2,3,3}}
                    {{2},{1,2}}    {{1,2,3,4}}        {{1,2,3,3,3}}
                    {{1},{1},{1}}  {{1},{1,1,1}}      {{1,2,3,4,4}}
                                   {{1,1},{1,1}}      {{1,2,3,4,5}}
                                   {{1},{1,2,2}}      {{1},{1,1,1,1}}
                                   {{1,2},{2,2}}      {{1,1},{1,1,1}}
                                   {{1,3},{2,3}}      {{1,1},{1,2,2}}
                                   {{2},{1,2,2}}      {{1},{1,2,2,2}}
                                   {{3},{1,2,3}}      {{1,2},{2,2,2}}
                                   {{1},{1},{1,1}}    {{1,2},{2,3,3}}
                                   {{1},{2},{1,2}}    {{1,3},{2,3,3}}
                                   {{2},{2},{1,2}}    {{1,4},{2,3,4}}
                                   {{1},{1},{1},{1}}  {{2},{1,1,2,2}}
                                                      {{2},{1,2,2,2}}
                                                      {{2},{1,2,3,3}}
                                                      {{2,2},{1,2,2}}
                                                      {{3},{1,2,3,3}}
                                                      {{3,3},{1,2,3}}
                                                      {{4},{1,2,3,4}}
                                                      {{1},{1},{1,1,1}}
                                                      {{1},{1,1},{1,1}}
                                                      {{1},{1},{1,2,2}}
                                                      {{1},{1,2},{2,2}}
                                                      {{1},{2},{1,2,2}}
                                                      {{2},{1,2},{2,2}}
                                                      {{2},{1,3},{2,3}}
                                                      {{2},{2},{1,2,2}}
                                                      {{2},{3},{1,2,3}}
                                                      {{3},{1,3},{2,3}}
                                                      {{3},{3},{1,2,3}}
                                                      {{1},{1},{1},{1,1}}
                                                      {{1},{2},{2},{1,2}}
                                                      {{2},{2},{2},{1,2}}
                                                      {{1},{1},{1},{1},{1}}

Crossrefs

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321227, A321228, A321231.

A322118 Number of non-isomorphic connected multiset partitions of weight n with no singletons that cannot be capped by a tree.

Original entry on oeis.org

1, 1, 2, 3, 7, 11, 29, 55, 155, 386, 1171
Offset: 0

Views

Author

Gus Wiseman, Nov 26 2018

Keywords

Comments

The density of a multiset partition is defined to be the sum of numbers of distinct elements in each part, minus the number of parts, minus the total number of distinct elements in the whole partition. A multiset partition is a tree if it has more than one part, is connected, and has density -1. A cap is a certain kind of non-transitive coarsening of a multiset partition. For example, the four caps of {{1,1},{1,2},{2,2}} are {{1,1},{1,2},{2,2}}, {{1,1},{1,2,2}}, {{1,1,2},{2,2}}, {{1,1,2,2}}. - Gus Wiseman, Feb 05 2021

Examples

			The multiset partition C = {{1,1},{1,2,3},{2,3,3}} is not a tree but has the cap {{1,1},{1,2,3,3}} which is a tree, so C is not counted under a(8).
Non-isomorphic representatives of the a(2) = 2 through a(6) = 29 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}    {{1,1,1,1,1,1}}
  {{1,2}}  {{1,2,2}}  {{1,1,2,2}}    {{1,1,2,2,2}}    {{1,1,1,2,2,2}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}    {{1,1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}    {{1,1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}    {{1,2,2,2,2,2}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}    {{1,2,2,3,3,3}}
                      {{1,2},{1,2}}  {{1,2,3,4,5}}    {{1,2,3,3,3,3}}
                                     {{1,1},{1,1,1}}  {{1,2,3,3,4,4}}
                                     {{1,2},{1,2,2}}  {{1,2,3,4,4,4}}
                                     {{2,2},{1,2,2}}  {{1,2,3,4,5,5}}
                                     {{2,3},{1,2,3}}  {{1,2,3,4,5,6}}
                                                      {{1,1},{1,1,1,1}}
                                                      {{1,1,1},{1,1,1}}
                                                      {{1,1,2},{1,2,2}}
                                                      {{1,2},{1,1,2,2}}
                                                      {{1,2},{1,2,2,2}}
                                                      {{1,2},{1,2,3,3}}
                                                      {{1,2,2},{1,2,2}}
                                                      {{1,2,3},{1,2,3}}
                                                      {{1,2,3},{2,3,3}}
                                                      {{1,3,4},{2,3,4}}
                                                      {{2,2},{1,1,2,2}}
                                                      {{2,2},{1,2,2,2}}
                                                      {{2,3},{1,2,3,3}}
                                                      {{3,3},{1,2,3,3}}
                                                      {{3,4},{1,2,3,4}}
                                                      {{1,1},{1,1},{1,1}}
                                                      {{1,2},{1,2},{1,2}}
                                                      {{1,2},{1,3},{2,3}}

Links

Crossrefs

Non-isomorphic tree multiset partitions are counted by A321229, or A321231 without singletons.
The version with singletons is A322110.
The weak-antichain case is counted by A322138, or A322117 with singletons.
Cf. A002218, A007718, A013922, A030019, A275307, A293994, A304118, A304382, A304887, A305079, A319719, A319721, A321194.

Extensions

Definition corrected by Gus Wiseman, Feb 05 2021

A321227 Number of connected multiset partitions with multiset density -1 of strongly normal multisets of size n.

Original entry on oeis.org

0, 1, 3, 6, 17, 43, 147, 458, 1729, 6445, 27011
Offset: 0

Views

Author

Gus Wiseman, Oct 31 2018

Keywords

Comments

The multiset density of a multiset partition is the sum of the numbers of distinct vertices in each part minus the number of parts minus the number of vertices.
A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition its multiplicities are weakly decreasing.

Examples

			The a(1) = 1 through a(4) = 17 multiset partitions:
  {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
         {{1,2}}    {{1,1,2}}      {{1,1,1,2}}
         {{1},{1}}  {{1,2,3}}      {{1,1,2,2}}
                    {{1},{1,1}}    {{1,1,2,3}}
                    {{1},{1,2}}    {{1,2,3,4}}
                    {{1},{1},{1}}  {{1},{1,1,1}}
                                   {{1,1},{1,1}}
                                   {{1},{1,1,2}}
                                   {{1,1},{1,2}}
                                   {{1},{1,2,2}}
                                   {{1},{1,2,3}}
                                   {{1,2},{1,3}}
                                   {{2},{1,1,2}}
                                   {{1},{1},{1,1}}
                                   {{1},{1},{1,2}}
                                   {{1},{2},{1,2}}
                                   {{1},{1},{1},{1}}

Crossrefs

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321228, A321229, A321231.

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]]]];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
mensity[c_]:=Total[(Length[Union[#]]-1&)/@c]-Length[Union@@c];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Sum[Length[Select[mps[m],And[mensity[#]==-1,Length[csm[#]]==1]&]],{m,strnorm[n]}],{n,0,8}]
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