A316980 -id:A316980 - cp's OEIS frontend

A323655 Number of non-isomorphic multiset partitions of weight n with at most 2 distinct vertices, or with at most 2 (not necessarily distinct) edges.

1, 1, 4, 7, 19, 35, 80, 149, 307, 566, 1092, 1974, 3643, 6447, 11498, 19947, 34636, 58974, 100182, 167713, 279659, 461056, 756562, 1230104, 1990255, 3195471, 5105540, 8103722, 12801925, 20107448, 31439978, 48907179, 75755094, 116797754, 179354540, 274253042

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Also the number of nonnegative integer matrices with only one or two columns, no zero rows or columns, and sum of entries equal to n, up to row and column permutations.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 distinct vertices:
  {{1}}  {{11}}    {{111}}      {{1111}}
         {{12}}    {{122}}      {{1122}}
         {{1}{1}}  {{1}{11}}    {{1222}}
         {{1}{2}}  {{1}{22}}    {{1}{111}}
                   {{2}{12}}    {{11}{11}}
                   {{1}{1}{1}}  {{1}{122}}
                   {{1}{2}{2}}  {{11}{22}}
                                {{12}{12}}
                                {{1}{222}}
                                {{12}{22}}
                                {{2}{122}}
                                {{1}{1}{11}}
                                {{1}{1}{22}}
                                {{1}{2}{12}}
                                {{1}{2}{22}}
                                {{2}{2}{12}}
                                {{1}{1}{1}{1}}
                                {{1}{1}{2}{2}}
                                {{1}{2}{2}{2}}
Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 edges:
  {{1}}  {{11}}    {{111}}    {{1111}}
         {{12}}    {{122}}    {{1122}}
         {{1}{1}}  {{123}}    {{1222}}
         {{1}{2}}  {{1}{11}}  {{1233}}
                   {{1}{22}}  {{1234}}
                   {{1}{23}}  {{1}{111}}
                   {{2}{12}}  {{11}{11}}
                              {{1}{122}}
                              {{11}{22}}
                              {{12}{12}}
                              {{1}{222}}
                              {{12}{22}}
                              {{1}{233}}
                              {{12}{33}}
                              {{1}{234}}
                              {{12}{34}}
                              {{13}{23}}
                              {{2}{122}}
                              {{3}{123}}
Inequivalent representatives of the a(4) = 19 matrices:
  [4] [2 2] [1 3]
.
  [1] [1 0] [1 0] [0 1] [2] [2 0] [1 1] [1 1]
  [3] [1 2] [0 3] [1 2] [2] [0 2] [1 1] [0 2]
.
  [1] [1 0] [1 0] [1 0] [0 1]
  [1] [1 0] [0 1] [0 1] [0 1]
  [2] [0 2] [1 1] [0 2] [1 1]
.
  [1] [1 0] [1 0]
  [1] [1 0] [0 1]
  [1] [0 1] [0 1]
  [1] [0 1] [0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A007716, A052847, A005380, A054974, A005986, A120733, A316980, A323654, A323655, A323656.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat(1, (EulerT(vector(n, k, k+1)) + EulerT(vector(n, k, if(k%2, 0, (k+6)\4))))/2)} \\ Andrew Howroyd, Aug 26 2019

a(2*n) = (A005380(2*n) + A005986(n))/2; a(2*n+1) = A005380(2*n+1)/2. - Andrew Howroyd, Aug 26 2019

Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019

A323656 Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.

Original entry on oeis.org

0, 0, 2, 4, 14, 28, 69, 134, 285, 536, 1050, 1918, 3566, 6346, 11363, 19771, 34405, 58677, 99797, 167223, 279032, 460264, 755560, 1228849, 1988680, 3193513, 5103104, 8100712, 12798207, 20102883, 31434374, 48900337, 75746745, 116787611, 179342230, 274238159

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Also the number of nonnegative integer matrices with only two columns, no zero rows or columns, and sum of entries equal to n, up to row and column permutations.

			Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 distinct vertices:
  {{12}}    {{122}}      {{1122}}
  {{1}{2}}  {{1}{22}}    {{1222}}
            {{2}{12}}    {{1}{122}}
            {{1}{2}{2}}  {{11}{22}}
                         {{12}{12}}
                         {{1}{222}}
                         {{12}{22}}
                         {{2}{122}}
                         {{1}{1}{22}}
                         {{1}{2}{12}}
                         {{1}{2}{22}}
                         {{2}{2}{12}}
                         {{1}{1}{2}{2}}
                         {{1}{2}{2}{2}}
Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 edges:
  {{1}{1}}  {{1}{11}}  {{1}{111}}
  {{1}{2}}  {{1}{22}}  {{11}{11}}
            {{1}{23}}  {{1}{122}}
            {{2}{12}}  {{11}{22}}
                       {{12}{12}}
                       {{1}{222}}
                       {{12}{22}}
                       {{1}{233}}
                       {{12}{33}}
                       {{1}{234}}
                       {{12}{34}}
                       {{13}{23}}
                       {{2}{122}}
                       {{3}{123}}
Inequivalent representatives of the a(4) = 14 matrices:
  [2 2] [1 3]
.
  [1 0] [1 0] [0 1] [2 0] [1 1] [1 1]
  [1 2] [0 3] [1 2] [0 2] [1 1] [0 2]
.
  [1 0] [1 0] [1 0] [0 1]
  [1 0] [0 1] [0 1] [0 1]
  [0 2] [1 1] [0 2] [1 1]
.
  [1 0] [1 0]
  [1 0] [0 1]
  [0 1] [0 1]
  [0 1] [0 1]

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000041, A007716, A052847, A054974, A120733, A316980, A323654, A323655, A323656.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat(0, (EulerT(vector(n, k, k+1)) + EulerT(vector(n, k, if(k%2, 0, (k+6)\4))))/2 - EulerT(vector(n,k,1)))} \\ Andrew Howroyd, Aug 26 2019

a(n) = A323655(n) - A000041(n). - Andrew Howroyd, Aug 26 2019

Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019

A319629 Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of distinct multisets.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 7, 9, 29, 66, 189

Offset: 0

Gus Wiseman, Sep 25 2018

The dual of a multiset partition has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.

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(7) = 9 antichains:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
4: {{1,1,1,1}}
5: {{1,1,1,1,1}}
   {{1,1},{1,2,2}}
6: {{1,1,1,1,1,1}}
   {{1,1},{1,2,2,2}}
   {{1,1,2},{1,2,2}}
   {{1,1,2},{2,2,2}}
   {{1,1,2},{2,3,3}}
   {{1,1},{1,2},{2,2}}
   {{1,2},{1,3},{2,3}}
7: {{1,1,1,1,1,1,1}}
   {{1,1},{1,2,2,2,2}}
   {{1,1,1},{1,2,2,2}}
   {{1,1,2},{1,2,2,2}}
   {{1,1,2},{2,2,2,2}}
   {{1,1,2},{2,3,3,3}}
   {{1,1},{1,2},{2,2,2}}
   {{1,1},{1,2},{2,3,3}}
   {{1,2},{1,3},{2,3,3}}

Cf. A006126, A007716, A007718, A056156, A059201, A283877, A316980, A316983, A318099, A319557, A319558, A319565, A319616-A319646.

Euler transform is A319644.

A319762 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 4, 9, 24

Offset: 0

Gus Wiseman, Sep 27 2018

A set multipartition is intersecting if no two parts are disjoint. The weight of a set multipartition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 1 through a(9) = 9 set multipartitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3,4},{2,3,4}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,4},{1,2,5},{3,4,5}}
   {{1,2},{1,3},{2,3},{2,3}}
9: {{1,3},{1,4,5},{2,3,4,5}}
   {{1,5},{1,6},{2,3,4,5,6}}
   {{2,5},{1,2,6},{3,4,5,6}}
   {{1,2,3},{2,4,5},{3,4,5}}
   {{1,3,5},{2,3,6},{4,5,6}}
   {{1,2},{1,3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1,3},{1,4},{1,4},{2,3,4}}
   {{1,3},{1,4},{3,4},{2,3,4}}

Cf. A007716, A049311, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.

Cf. A319752, A319759, A319763, A319764, A319765, A319779, A319781.

A319763 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 2, 12, 46, 181

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset partition is intersecting if no two parts are disjoint. 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(6) = 1 through a(8) = 12 multiset partitions:
6: {{1,2},{1,3},{2,3}}
7: {{1,2},{1,3},{2,3,3}}
   {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3},{2,2,3,3}}
   {{1,2},{1,3},{2,3,3,3}}
   {{1,2},{1,3},{2,3,4,4}}
   {{1,2},{1,3,3},{2,3,3}}
   {{1,2},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4,4}}
   {{1,3},{1,1,2},{2,3,3}}
   {{1,3},{1,2,2},{2,3,3}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,3},{1,2,4},{3,4,4}}
   {{2,4},{1,2,3},{3,4,4}}
   {{2,4},{1,2,5},{3,4,5}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317757, A318715, A318717.

Cf. A319752, A319759, A319762, A319764, A319765, A319779, A319781.

A319764 Number of non-isomorphic intersecting set systems of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 3, 8, 18

Offset: 0

Gus Wiseman, Sep 27 2018

A set system is a finite set of finite nonempty sets. It is intersecting if no two parts are disjoint. The weight of a set system is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(6) = 1 through a(9) = 8 set systems:
6: {{1,2},{1,3},{2,3}}
7: {{1,3},{1,4},{2,3,4}}
8: {{1,2},{1,3,4},{2,3,4}}
   {{1,4},{1,5},{2,3,4,5}}
   {{2,4},{1,2,5},{3,4,5}}
9: {{1,3},{1,4,5},{2,3,4,5}}
   {{1,5},{1,6},{2,3,4,5,6}}
   {{2,5},{1,2,6},{3,4,5,6}}
   {{1,2,3},{2,4,5},{3,4,5}}
   {{1,3,5},{2,3,6},{4,5,6}}
   {{1,2},{1,3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3},{1,2,3}}
   {{1,3},{1,4},{3,4},{2,3,4}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A317752, A317755, A317757, A318715, A318717.

Cf. A319752, A319759, A319762, A319763, A319765, A319779, A319781.

A319775 Number of non-isomorphic multiset partitions of weight n with empty intersection and no part containing all the vertices.

Original entry on oeis.org

1, 0, 1, 4, 16, 52, 185, 625, 2226, 7840, 28405

Offset: 0

Gus Wiseman, Sep 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) = 1 through a(4) = 16 multiset partitions:
2: {{1},{2}}
3: {{1},{2,2}}
   {{1},{2,3}}
   {{1},{2},{2}}
   {{1},{2},{3}}
4: {{1},{2,2,2}}
   {{1},{2,3,3}}
   {{1},{2,3,4}}
   {{1,1},{2,2}}
   {{1,2},{3,3}}
   {{1,2},{3,4}}
   {{1},{1},{2,2}}
   {{1},{1},{2,3}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{2},{3,4}}
   {{1},{3},{2,3}}
   {{1},{1},{2},{2}}
   {{1},{2},{2},{2}}
   {{1},{2},{3},{3}}
   {{1},{2},{3},{4}}

Cf. A007716, A281116, A283877, A316980, A317752, A317755, A317757.

Cf. A319778, A319779, A319781, A319783.

A319782 Number of non-isomorphic intersecting strict T_0 multiset partitions of weight n.

Original entry on oeis.org

1, 1, 1, 4, 7, 17, 42, 98, 248, 631, 1657

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset partition is intersecting iff no two parts are disjoint. The weight of a multiset partition is the sum of sizes of its parts. 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}}. The T_0 condition means the dual is strict.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 7 multiset partitions:
1: {{1}}
2: {{1,1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1},{1,1}}
   {{2},{1,2}}
4: {{1,1,1,1}}
   {{1,2,2,2}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}

Cf. A007716, A049311, A059201, A281116, A283877, A305843, A305854, A306006, A316980.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319789.

A319783 Number of set systems spanning n vertices with empty intersection whose dual is also a set system with empty intersection.

Original entry on oeis.org

1, 0, 0, 1, 203, 490572

Offset: 0

Gus Wiseman, Sep 27 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}}.

			The a(3) = 1 set system is {{1,2},{1,3},{2,3}}.

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757.

Cf. A319775, A319778, A319779, A319781.

A321588 Number of connected nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.

Original entry on oeis.org

1, 1, 1, 9, 29, 181, 1285, 10635, 102355, 1118021, 13637175, 184238115, 2727293893, 43920009785, 764389610843, 14297306352937, 286014489487815, 6093615729757841, 137750602009548533, 3293082026520294529, 83006675263513350581, 2200216851785981586729, 61180266502369886181253

Offset: 0

Gus Wiseman, Nov 13 2018

nonn

A matrix is connected if the positions in each row (or each column) of the nonzero entries form a connected hypergraph.

			The a(4) = 29 matrices:
4 31 13
.
3 21 21 20 12 12 11 110 11 110 101 101 1 10 10 02 011 011 01 01
1 10 01 11 10 01 20 101 02 011 110 011 3 21 12 11 110 101 21 12
.
11 11 10 10 01 01
10 01 11 01 11 10
01 10 01 11 10 11

Andrew Howroyd, Table of n, a(n) for n = 0..40

Cf. A007718, A056156, A059201, A120733, A283877, A316980, A319557, A319558, A319559, A319565, A319647, A319616-A319629, A321446, A321515, A369285.

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,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]]]]]]]]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@prs2mat[#],UnsameQ@@Transpose[prs2mat[#]],Length[csm[Map[Last,GatherBy[#,First],{2}]]]==1]&]],{n,6}]

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q,t,wf)={prod(j=1, #q, wf(t*q[j]))-1}
Q(m,n,wf=w->2)={my(s=0); forpart(p=m, s+=(-1)^#p*permcount(p)*exp(-sum(t=1, n, (-1)^t*x^t*K(p,t,wf)/t, O(x*x^n))) ); Vec((-1)^m*serchop(serlaplace(s),1), -n)}
ConnectedMats(M)={my([m, n]=matsize(M), R=matrix(m, n)); for(m=1, m, for(n=1, n, R[m, n] = M[m, n] - sum(i=1, m-1, sum(j=1, n-1, binomial(m-1, i-1)*binomial(n, j)*R[i, j]*M[m-i, n-j])))); R}
seq(n)={my(R=vectorv(n,m,Q(m,n,w->1/(1 - y^w) + O(y*y^n)))); for(i=2, #R, R[i] -= i*R[i-1]); Vec(1 + vecsum( vecsum( Vec( ConnectedMats( Mat(R))))))} \\ Andrew Howroyd, Jan 24 2024

a(7) onwards from Andrew Howroyd, Jan 24 2024

cp's OEIS Frontend

A323655 Number of non-isomorphic multiset partitions of weight n with at most 2 distinct vertices, or with at most 2 (not necessarily distinct) edges.

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A323656 Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.

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A319629 Number of non-isomorphic connected weight-n antichains of distinct multisets whose dual is also an antichain of distinct multisets.

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A319762 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n with empty intersection.

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A319763 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n with empty intersection.

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A319764 Number of non-isomorphic intersecting set systems of weight n with empty intersection.

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A319775 Number of non-isomorphic multiset partitions of weight n with empty intersection and no part containing all the vertices.

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A319782 Number of non-isomorphic intersecting strict T_0 multiset partitions of weight n.

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A319783 Number of set systems spanning n vertices with empty intersection whose dual is also a set system with empty intersection.

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A321588 Number of connected nonnegative integer matrices with sum of entries equal to n, no zero rows or columns, and distinct rows and columns.

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