A322111 -id:A322111 - cp's OEIS frontend

A320798 Number of non-isomorphic weight-n connected antichains of non-constant multisets with multiset density -1.

0, 1, 2, 5, 9, 24, 51, 134, 328, 868

Offset: 1

Gus Wiseman, Nov 29 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 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(6) = 24 multiset partitions:
  {{12}}  {{122}}  {{1122}}    {{11222}}    {{111222}}
          {{123}}  {{1222}}    {{12222}}    {{112222}}
                   {{1233}}    {{12233}}    {{112233}}
                   {{1234}}    {{12333}}    {{122222}}
                   {{13}{23}}  {{12344}}    {{122333}}
                               {{12345}}    {{123333}}
                               {{12}{233}}  {{123344}}
                               {{13}{233}}  {{123444}}
                               {{14}{234}}  {{123455}}
                                            {{123456}}
                                            {{112}{233}}
                                            {{122}{233}}
                                            {{12}{2333}}
                                            {{123}{344}}
                                            {{124}{344}}
                                            {{125}{345}}
                                            {{13}{2233}}
                                            {{13}{2333}}
                                            {{13}{2344}}
                                            {{133}{233}}
                                            {{14}{2344}}
                                            {{15}{2345}}
                                            {{13}{24}{34}}
                                            {{14}{24}{34}}

Cf. A007716, A007718, A286520, A304867, A319719, A319721, A321155, A321760, A322111, A322113.

A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 35, 85, 231, 633, 1845, 5461, 16707, 51945, 164695, 529077, 1722279, 5664794, 18813369, 62996850, 212533226, 721792761, 2466135375, 8471967938, 29249059293, 101440962296, 353289339927, 1235154230060, 4333718587353, 15255879756033

Offset: 0

Christian G. Bower, Nov 15 1998

nonn

Also the number of non-isomorphic connected multigraphs with loops with n edges and multiset density -1, where 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. - Gus Wiseman, Nov 28 2018

Alois P. Heinz, Table of n, a(n) for n = 0..1717
Richard J. Mathar, Counting Connected Graphs without Overlapping Cycles, arXiv:1808.06264 [math.CO], 2018.
Gus Wiseman, Non-isomorphic representatives of the a(1) = 2 through a(5) = 35 connected multigraphs with loops with multiset density -1.
Index entries for sequences related to trees

Essentially the same as A036251.

Cf. A000055, A007718, A007719, A038052, A191646, A303837, A321155, A321229, A321254, A321256, A322111.

max = 30; B[] = 1; Do[B[x] = x*Exp[Sum[(B[x^k] + x^k)/k + O[x]^n, {k, 1, n}]] // Normal, {n, 1, max}]; A[x_] = B[x] - B[x]^2/2 + B[x^2]/2; CoefficientList[1 + A[x] + O[x]^max, x] (* Jean-François Alcover, Jan 28 2019 *)

G.f.: B(x) - B^2(x)/2 + B(x^2)/2, where B(x) is g.f. for A036249.

A322113 Number of non-isomorphic self-dual connected antichains of multisets of weight n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 3, 5, 10, 18, 30

Offset: 0

Gus Wiseman, Nov 26 2018

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(9) = 18 antichains:
  {{1}}  {{11}}  {{111}}  {{1111}}    {{11111}}    {{111111}}
                          {{12}{12}}  {{11}{122}}  {{112}{122}}
                                                   {{12}{13}{23}}
.
  {{1111111}}      {{11111111}}        {{111111111}}
  {{111}{1222}}    {{111}{11222}}      {{1111}{12222}}
  {{112}{1222}}    {{1112}{1222}}      {{1112}{11222}}
  {{11}{12}{233}}  {{112}{12222}}      {{1112}{12222}}
  {{12}{13}{233}}  {{1122}{1122}}      {{112}{122222}}
                   {{11}{122}{233}}    {{11}{11}{12233}}
                   {{12}{13}{2333}}    {{11}{122}{1233}}
                   {{13}{112}{233}}    {{112}{123}{233}}
                   {{13}{122}{233}}    {{113}{122}{233}}
                   {{12}{13}{24}{34}}  {{12}{111}{2333}}
                                       {{12}{13}{23333}}
                                       {{12}{133}{2233}}
                                       {{123}{123}{123}}
                                       {{13}{112}{2333}}
                                       {{22}{113}{2333}}
                                       {{12}{13}{14}{234}}
                                       {{12}{13}{22}{344}}
                                       {{12}{13}{24}{344}}

Cf. A006126, A007716, A007718, A286520, A293993, A293994, A304867, A316983, A318099, A319719, A319721, A322111, A322112.

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

A320798 Number of non-isomorphic weight-n connected antichains of non-constant multisets with multiset density -1.

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A036250 Number of trees of nonempty sets with n points. (Each node is a set of 1 or more points.)

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A322113 Number of non-isomorphic self-dual connected antichains of multisets of weight n.

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

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