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 1-6 of 6 results.

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.

A321253 Number of non-isomorphic strict connected weight-n multiset partitions with multiset density -1.

Original entry on oeis.org

0, 1, 2, 5, 12, 28, 78, 202, 578, 1650, 4904
Offset: 0

Views

Author

Gus Wiseman, Nov 01 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) = 28 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}}
                  {{1,2,3}}    {{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}}    {{1,2,3,4,4}}
                               {{1},{1,2,2}}    {{1,2,3,4,5}}
                               {{1,2},{2,2}}    {{1},{1,1,1,1}}
                               {{1,3},{2,3}}    {{1,1},{1,1,1}}
                               {{2},{1,2,2}}    {{1,1},{1,2,2}}
                               {{3},{1,2,3}}    {{1},{1,2,2,2}}
                               {{1},{2},{1,2}}  {{1,2},{2,2,2}}
                                                {{1,2},{2,3,3}}
                                                {{1,3},{2,3,3}}
                                                {{1,4},{2,3,4}}
                                                {{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},{1,2},{2,2}}
                                                {{1},{2},{1,2,2}}
                                                {{2},{1,2},{2,2}}
                                                {{2},{1,3},{2,3}}
                                                {{2},{3},{1,2,3}}
                                                {{3},{1,3},{2,3}}

Crossrefs

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A318697, A321155, A321194, A321228, A321229, A321254.

A321231 Number of non-isomorphic connected weight-n multiset partitions with no singletons and multiset density -1.

Original entry on oeis.org

1, 0, 2, 3, 8, 15, 42, 94, 256, 656, 1807
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(2) = 2 through a(5) = 15 multiset partitions:
  {{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}}
           {{1,2,3}}  {{1,2,2,2}}    {{1,2,2,2,2}}
                      {{1,2,3,3}}    {{1,2,2,3,3}}
                      {{1,2,3,4}}    {{1,2,3,3,3}}
                      {{1,1},{1,1}}  {{1,2,3,4,4}}
                      {{1,2},{2,2}}  {{1,2,3,4,5}}
                      {{1,3},{2,3}}  {{1,1},{1,1,1}}
                                     {{1,1},{1,2,2}}
                                     {{1,2},{2,2,2}}
                                     {{1,2},{2,3,3}}
                                     {{1,3},{2,3,3}}
                                     {{1,4},{2,3,4}}
                                     {{2,2},{1,2,2}}
                                     {{3,3},{1,2,3}}

Crossrefs

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A317672, A318697, A321155, A321228, A321229.

A321255 Number of connected multiset partitions with multiset density -1, of strongly normal multisets of size n, with no singletons.

Original entry on oeis.org

0, 0, 2, 3, 8, 19, 60, 183, 643, 2355, 9393
Offset: 0

Views

Author

Gus Wiseman, Nov 01 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(2) = 2 through a(5) = 19 multiset partitions:
  {{1,1}}  {{1,1,1}}  {{1,1,1,1}}    {{1,1,1,1,1}}
  {{1,2}}  {{1,1,2}}  {{1,1,1,2}}    {{1,1,1,1,2}}
           {{1,2,3}}  {{1,1,2,2}}    {{1,1,1,2,2}}
                      {{1,1,2,3}}    {{1,1,1,2,3}}
                      {{1,2,3,4}}    {{1,1,2,2,3}}
                      {{1,1},{1,1}}  {{1,1,2,3,4}}
                      {{1,1},{1,2}}  {{1,2,3,4,5}}
                      {{1,2},{1,3}}  {{1,1},{1,1,1}}
                                     {{1,1},{1,1,2}}
                                     {{1,1},{1,2,2}}
                                     {{1,1},{1,2,3}}
                                     {{1,2},{1,1,1}}
                                     {{1,2},{1,1,3}}
                                     {{1,2},{1,3,4}}
                                     {{1,3},{1,1,2}}
                                     {{1,3},{1,2,2}}
                                     {{1,3},{1,2,4}}
                                     {{1,4},{1,2,3}}
                                     {{2,3},{1,1,2}}

Crossrefs

Cf. A000272, A030019, A052888, A134954, A304867, A317672, A321228, A321229, A321253.

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}]

A321256 Regular triangle where T(n,k) is the number of non-isomorphic connected set systems of weight n with density -1 <= k <= n-2.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 4, 0, 0, 0, 6, 1, 0, 0, 0, 13, 5, 0, 0, 0, 0, 23, 12, 2, 0, 0, 0, 0, 49, 36, 11, 0, 0, 0, 0, 0, 100, 95, 39, 5, 0, 0, 0, 0, 0, 220, 262, 143, 32, 1, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Gus Wiseman, Nov 01 2018

Keywords

Comments

A set system is a finite set of finite nonempty sets. The density of a set system is the sum of sizes of each part (weight) minus the number of parts minus the number of vertices.

Examples

			Triangle begins:
    1
    1   0
    2   0   0
    4   0   0   0
    6   1   0   0   0
   13   5   0   0   0   0
   23  12   2   0   0   0   0
   49  36  11   0   0   0   0   0
  100  95  39   5   0   0   0   0   0
  220 262 143  32   1   0   0   0   0   0

Crossrefs

First column is A321228. Row sums are A007718.
Cf. A007716, A030019, A056156, A283877, A293607, A300913, A319557, A321155, A321253, A321254.
Showing 1-6 of 6 results.

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