A304887 -id:A304887 - cp's OEIS frontend

A322139 Number of labeled 2-connected simple graphs with n edges (the vertices are {1,2,...,k} for some k).

1, 1, 0, 1, 3, 18, 131, 1180, 12570, 154535, 2151439, 33431046, 573197723, 10743619285, 218447494812, 4787255999220, 112454930390211, 2818138438707516, 75031660452368001, 2114705500316025737, 62890323682634277951, 1967901134191778583146, 64623905086814216468839

Offset: 0

Gus Wiseman, Nov 27 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..100
Gus Wiseman, The a(6) = 131 labeled 2-connected graphs with 6 edges.

Cf. A002218, A013922, A123534, A275307, A291841, A304118, A304887, A322110, A322117, A322118, A322137, A322138.

seq(n)={Vec(1 + vecsum(Vec(serlaplace(log(x/serreverse(x*deriv(log(sum(k=0, n, (1 + y + O(y*y^n))^binomial(k, 2) * x^k / k!) + O(x*x^n)))))))))} \\ Andrew Howroyd, Nov 29 2018

a(n) = Sum_{i=3..n} A123534(i, n). - Andrew Howroyd, Nov 30 2018

Terms a(7) and beyond from Andrew Howroyd, Nov 29 2018

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

Gus Wiseman, Oct 31 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) = 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}}

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

A322138 Number of non-isomorphic weight-n blobs (2-connected weak antichains) of multisets with no singletons.

Original entry on oeis.org

1, 0, 2, 3, 7, 7, 20, 26, 78, 184, 553

Offset: 0

Gus Wiseman, Nov 27 2018

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) = 2 through a(7) = 26 blobs:
  {{11}}  {{111}}  {{1111}}    {{11111}}  {{111111}}      {{1111111}}
  {{12}}  {{122}}  {{1122}}    {{11222}}  {{111222}}      {{1112222}}
          {{123}}  {{1222}}    {{12222}}  {{112222}}      {{1122222}}
                   {{1233}}    {{12233}}  {{112233}}      {{1122333}}
                   {{1234}}    {{12333}}  {{122222}}      {{1222222}}
                   {{11}{11}}  {{12344}}  {{122333}}      {{1222333}}
                   {{12}{12}}  {{12345}}  {{123333}}      {{1223333}}
                                          {{123344}}      {{1223344}}
                                          {{123444}}      {{1233333}}
                                          {{123455}}      {{1233444}}
                                          {{123456}}      {{1234444}}
                                          {{111}{111}}    {{1234455}}
                                          {{112}{122}}    {{1234555}}
                                          {{122}{122}}    {{1234566}}
                                          {{123}{123}}    {{1234567}}
                                          {{123}{233}}    {{112}{1222}}
                                          {{134}{234}}    {{122}{1233}}
                                          {{11}{11}{11}}  {{123}{2233}}
                                          {{12}{12}{12}}  {{123}{2333}}
                                          {{12}{13}{23}}  {{123}{2344}}
                                                          {{134}{2344}}
                                                          {{145}{2345}}
                                                          {{223}{1233}}
                                                          {{344}{1234}}
                                                          {{12}{13}{233}}
                                                          {{13}{14}{234}}

Cf. A002218, A007718, A013922, A275307, A286520, A293994, A304118, A304887, A319719, A322110, A322117, A322118.

A305195 Number of z-blobs summing to n. Number of connected strict integer partitions of n, with pairwise indivisible parts, that cannot be capped by a z-tree.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 2, 1, 1, 1, 3, 2, 2, 2, 1, 1, 3, 3, 3, 1, 1, 1, 4, 5, 6, 2, 1, 1, 4, 6, 7, 2, 2, 6

Offset: 1

Gus Wiseman, May 27 2018

nonn

Caps of a clutter are defined in the link, and the generalization to "multiclutters," where edges can be multisets, is straightforward.

			The a(30) = 2 z-blobs together with the corresponding multiset systems:
     (30): {{1,2,3}}
  (18,12): {{1,2,2},{1,1,2}}
The a(47) = 3 z-blobs together with the corresponding multiset systems:
        (47): {{15}}
  (21,14,12): {{2,4},{1,4},{1,1,2}}
  (20,15,12): {{1,1,3},{2,3},{1,1,2}}
The a(60) = 5 z-blobs together with the corresponding multiset systems:
           (60): {{1,1,2,3}}
        (42,18): {{1,2,4},{1,2,2}}
        (36,24): {{1,1,2,2},{1,1,1,2}}
     (30,18,12): {{1,2,3},{1,2,2},{1,1,2}}
  (21,15,14,10): {{2,4},{2,3},{1,4},{1,3}}
The a(67) = 7 z-blobs together with the corresponding multiset systems:
           (67): {{19}}
     (45,12,10): {{2,2,3},{1,1,2},{1,3}}
     (42,15,10): {{1,2,4},{2,3},{1,3}}
     (40,15,12): {{1,1,1,3},{2,3},{1,1,2}}
     (33,22,12): {{2,5},{1,5},{1,1,2}}
     (28,21,18): {{1,1,4},{2,4},{1,2,2}}
  (24,18,15,10): {{1,1,1,2},{1,2,2},{2,3},{1,3}}

Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.

Cf. A030019, A048143, A275307, A285572, A293510, A303362, A303837, A303838, A304118, A304382, A304887, A305028, A305078, A305194.

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

Gus Wiseman, Oct 31 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.

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

			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}}

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

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}]

A322140 Number of labeled 2-connected multigraphs with n edges (the vertices are {1,2,...,k} for some k).

Original entry on oeis.org

1, 1, 1, 2, 7, 37, 262, 2312, 24338, 296928, 4112957, 63692909, 1089526922, 20389411551, 414146189901, 9070116944468, 212983762029683, 5336570227705763, 142083405456873290, 4004953714929148655, 119128974685786590410, 3728639072095285867881

Offset: 0

Gus Wiseman, Nov 27 2018

nonn

We consider a single edge to be 2-connected, so a(1) = 1.

Andrew Howroyd, Table of n, a(n) for n = 0..100
Gus Wiseman, The a(5) = 37 labeled 2-connected multigraphs with 5 edges.

Cf. A002218, A002905, A013922, A275307, A291841, A304118, A304887, A322110, A322117, A322118, A322137, A322138.

seq(n)={Vec(1 + vecsum(Vec(serlaplace(log(x/serreverse(x*deriv(log(sum(k=0, n, 1/(1 - y + O(y*y^n))^binomial(k, 2) * x^k / k!) + O(x*x^n)))))))))} \\ Andrew Howroyd, Nov 29 2018

Terms a(7) and beyond from Andrew Howroyd, Nov 29 2018

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.

A322399 Number of non-isomorphic 2-edge-connected clutters spanning n vertices.

Original entry on oeis.org

0, 0, 2, 12, 149

Offset: 1

Gus Wiseman, Dec 06 2018

A clutter is a connected antichain of sets. It is 2-edge-connected if it cannot be disconnected by removing any single edge. Compare to blobs or 2-vertex-connected clutters (A304887).

			Non-isomorphic representatives of the a(4) = 12 clutters:
  {{1,4},{2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Cf. A002218, A007718, A013922, A030019, A048143, A095983, A304887, A317634, A317635, A317672, A322335, A322394, A322396, A322397.

cp's OEIS Frontend

A322139 Number of labeled 2-connected simple graphs with n edges (the vertices are {1,2,...,k} for some k).

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

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A322138 Number of non-isomorphic weight-n blobs (2-connected weak antichains) of multisets with no singletons.

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A305195 Number of z-blobs summing to n. Number of connected strict integer partitions of n, with pairwise indivisible parts, that cannot be capped by a z-tree.

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

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Mathematica

A322140 Number of labeled 2-connected multigraphs with n edges (the vertices are {1,2,...,k} for some k).

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Author

Keywords

Comments

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Programs

PARI

Extensions

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

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A322399 Number of non-isomorphic 2-edge-connected clutters spanning n vertices.

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