A321255 -id:A321255 - cp's OEIS frontend

A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

1, 3, 0, 6, 0, 0, 16, 1, 0, 0, 37, 3, 0, 0, 0, 105, 18, 2, 0, 0, 0, 279, 68, 7, 0, 0, 0, 0, 817, 293, 46, 3, 0, 0, 0, 0, 2387, 1141, 228, 17, 1, 0, 0, 0, 0, 7269, 4511, 1189, 135, 9, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Nov 01 2018

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.

			Triangle begins:
     1
     3    0
     6    0    0
    16    1    0    0
    37    3    0    0    0
   105   18    2    0    0    0
   279   68    7    0    0    0    0
   817  293   46    3    0    0    0    0
  2387 1141  228   17    1    0    0    0    0
  7269 4511 1189  135    9    0    0    0    0    0

First column is A321229. Row sums are A007718.

Cf. A007716, A317672, A321155, A321194, A321253, A321255, A321256.

A322111 Number of non-isomorphic self-dual connected multiset partitions of weight n with multiset density -1.

Original entry on oeis.org

1, 1, 1, 2, 2, 5, 5, 13, 13, 37, 37

Offset: 0

Gus Wiseman, Nov 26 2018

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 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}}. A multiset partition is self-dual if it is isomorphic to its dual. For example, {{1,1},{1,2,2},{2,3,3}} is self-dual, as it is isomorphic to its dual {{1,1,2},{2,2,3},{3,3}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

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

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A316983, A318697, A319616, A321155, A321255.

A322112 Number of non-isomorphic self-dual connected multiset partitions of weight n with no singletons and multiset density -1.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 4, 4, 9, 9

Offset: 0

Gus Wiseman, Nov 26 2018

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 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}}. A multiset partition is self-dual if it is isomorphic to its dual. For example, {{1,1},{1,2,2},{2,3,3}} is self-dual, as it is isomorphic to its dual {{1,1,2},{2,2,3},{3,3}}.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(10) = 9 multiset partitions:
  {{11}}  {{111}}  {{1111}}  {{11111}}    {{111111}}    {{1111111}}
                             {{11}{122}}  {{22}{1122}}  {{111}{1222}}
                                                        {{22}{11222}}
                                                        {{11}{12}{233}}
.
  {{11111111}}      {{111111111}}        {{1111111111}}
  {{111}{11222}}    {{1111}{12222}}      {{1111}{112222}}
  {{22}{112222}}    {{22}{1122222}}      {{22}{11222222}}
  {{11}{122}{233}}  {{222}{111222}}      {{222}{1112222}}
                    {{11}{11}{12233}}    {{111}{122}{2333}}
                    {{11}{113}{2233}}    {{22}{113}{23333}}
                    {{12}{111}{2333}}    {{22}{1133}{2233}}
                    {{22}{113}{2333}}    {{33}{33}{112233}}
                    {{12}{13}{22}{344}}  {{11}{14}{223}{344}}

Cf. A000272, A007716, A007718, A030019, A052888, A134954, A304867, A304887, A316983, A321155, A321255, A322111.

cp's OEIS Frontend

A321254 Regular triangle where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with multiset density -1 <= k <= n-2.

Views

Author

Keywords

Comments

Examples

Crossrefs

A322111 Number of non-isomorphic self-dual connected multiset partitions of weight n with multiset density -1.

Views

Author

Keywords

Comments

Examples

Crossrefs

A322112 Number of non-isomorphic self-dual connected multiset partitions of weight n with no singletons and multiset density -1.

Views

Author

Keywords

Comments

Examples

Crossrefs