A316980 -id:A316980 - cp's OEIS frontend

A323787 Number of non-isomorphic multiset partitions of strict multiset partitions of weight n.

1, 1, 4, 14, 56, 219, 1001, 4588

Offset: 0

Gus Wiseman, Jan 27 2019

The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 14 multiset partitions:
  {{1}}  {{11}}      {{111}}
         {{12}}      {{112}}
         {{1}{2}}    {{123}}
         {{1}}{{2}}  {{1}{11}}
                     {{1}{12}}
                     {{1}{23}}
                     {{2}{11}}
                     {{1}}{{11}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{2}}{{11}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}

Cf. A002846, A005121, A007716, A050343, A213427, A269134, A283877, A306186, A316980, A317791, A318564, A318565, A318566, A318812.

Cf. A323788, A323789, A323790, A323791, A323792, A323793, A323794, A323795.

A319779 Number of intersecting multiset partitions of weight n whose dual is not an intersecting multiset partition.

Original entry on oeis.org

1, 0, 0, 0, 1, 4, 20, 66, 226, 696, 2156

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 weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

			Non-isomorphic representatives of the a(4) = 1 through a(6) = 20 multiset partitions:
4: {{1,3},{2,3}}
5: {{1,2},{2,3,3}}
   {{1,3},{2,3,3}}
   {{1,4},{2,3,4}}
   {{3},{1,3},{2,3}}
6: {{1,2},{2,3,3,3}}
   {{1,3},{2,2,3,3}}
   {{1,3},{2,3,3,3}}
   {{1,3},{2,3,4,4}}
   {{1,4},{2,3,4,4}}
   {{1,5},{2,3,4,5}}
   {{1,1,2},{2,3,3}}
   {{1,2,2},{2,3,3}}
   {{1,2,3},{3,4,4}}
   {{1,2,4},{3,4,4}}
   {{1,2,5},{3,4,5}}
   {{1,3,3},{2,3,3}}
   {{1,3,4},{2,3,4}}
   {{2},{1,2},{2,3,3}}
   {{3},{1,3},{2,3,3}}
   {{4},{1,4},{2,3,4}}
   {{1,3},{2,3},{2,3}}
   {{1,3},{2,3},{3,3}}
   {{1,4},{2,4},{3,4}}
   {{3},{3},{1,3},{2,3}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319775, A319778, A319781, A319783.

A319781 Number of multiset partitions of integer partitions of n with empty intersection. Number of relatively prime factorizations of Heinz numbers of integer partitions of n.

Original entry on oeis.org

1, 0, 0, 1, 3, 9, 21, 48, 103, 214, 436, 863, 1689

Offset: 0

Gus Wiseman, Sep 27 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

			The a(3) = 1 through a(5) = 9 multiset partitions:
3: {{1},{2}}
4: {{1},{3}}
   {{2},{1,1}}
   {{1},{1},{2}}
5: {{1},{4}}
   {{2},{3}}
   {{3},{1,1}}
   {{1},{2,2}}
   {{1},{1},{3}}
   {{1},{2},{2}}
   {{2},{1,1,1}}
   {{1},{2},{1,1}}
   {{1},{1},{1},{2}}

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A318715.

Cf. A319775, A319779, A319778, A319783.

A319755 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 19, 30, 60, 107, 212

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(1) = 1 through a(5) = 9 set multipartitions:
1: {{1}}
2: {{1,2}}
   {{1},{1}}
3: {{1,2,3}}
   {{2},{1,2}}
   {{1},{1},{1}}
4: {{1,2,3,4}}
   {{3},{1,2,3}}
   {{1,2},{1,2}}
   {{1,3},{2,3}}
   {{2},{2},{1,2}}
   {{1},{1},{1},{1}}
5: {{1,2,3,4,5}}
   {{4},{1,2,3,4}}
   {{1,4},{2,3,4}}
   {{2,3},{1,2,3}}
   {{2},{1,2},{1,2}}
   {{3},{3},{1,2,3}}
   {{3},{1,3},{2,3}}
   {{2},{2},{2},{1,2}}
   {{1},{1},{1},{1},{1}}

Cf. A007716, A049311, A283877, A305854, A306006, A316980, A319616.

Cf. A319748, A319752, A319759, A319760, A319765, A319779, A319787, A319789.

A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.

Original entry on oeis.org

1, 1, 2, 14, 814, 1174774, 909125058112, 291200434263385001951232

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}}.

A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.

			The a(3) = 14 set systems:
   {{1},{1,2},{1,2,3}}
   {{1},{1,3},{1,2,3}}
   {{2},{1,2},{1,2,3}}
   {{2},{2,3},{1,2,3}}
   {{3},{1,3},{1,2,3}}
   {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,2},{1,3},{1,2,3}}
   {{1,2},{2,3},{1,2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{1},{1,2},{1,3},{1,2,3}}
   {{2},{1,2},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}

Cf. A007716, A281116, A283877, A305854, A306006,  A316980, A316983, A317757, A319616.
Cf. A319752, A319765, A319766, A319767, A319768, A319769.
Intersecting set-systems are A051185.
The unlabeled multiset partition version is A319773.
The covering case is A327037.
The version without strict dual is A327038.
Cointersecting set-systems are A327039.
The BII-numbers of these set-systems are A327061.
Cf. A003465, A305843, A305844, A326854, A327020, A327040, A327052.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Subsets[Subsets[Range[n],{1,n}]],Union@@#==Range[n]&&UnsameQ@@dual[#]&&stableQ[#,Intersection[#1,#2]=={}&]&&stableQ[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}] (* Gus Wiseman, Aug 19 2019 *)

a(6)-a(7) from Christian Sievers, Aug 18 2024

A340600 Number of non-isomorphic balanced multiset partitions of weight n.

Original entry on oeis.org

1, 1, 0, 4, 7, 16, 52, 206, 444, 1624, 5462, 19188, 62890, 215367, 765694, 2854202, 10634247, 39842786, 150669765, 581189458, 2287298588, 9157598354, 37109364812, 151970862472, 629048449881, 2635589433705, 11184718653563, 48064965080106, 208988724514022, 918639253237646, 4079974951494828

Offset: 0

Gus Wiseman, Feb 05 2021

nonn

We define a multiset partition to be balanced if it has exactly as many parts as the greatest size of a part.

			Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions (empty column indicated by dot):
  {{1}}  .  {{1},{1,1}}  {{1,1},{1,1}}  {{1},{1},{1,1,1}}
            {{1},{2,2}}  {{1,1},{2,2}}  {{1},{1},{1,2,2}}
            {{1},{2,3}}  {{1,2},{1,2}}  {{1},{1},{2,2,2}}
            {{2},{1,2}}  {{1,2},{2,2}}  {{1},{1},{2,3,3}}
                         {{1,2},{3,3}}  {{1},{1},{2,3,4}}
                         {{1,2},{3,4}}  {{1},{2},{1,2,2}}
                         {{1,3},{2,3}}  {{1},{2},{2,2,2}}
                                        {{1},{2},{2,3,3}}
                                        {{1},{2},{3,3,3}}
                                        {{1},{2},{3,4,4}}
                                        {{1},{2},{3,4,5}}
                                        {{1},{3},{2,3,3}}
                                        {{1},{4},{2,3,4}}
                                        {{2},{2},{1,2,2}}
                                        {{2},{3},{1,2,3}}
                                        {{3},{3},{1,2,3}}

The version for partitions is A047993.
The co-balanced version is A319616.
The cross-balanced version is A340651.
The twice-balanced version is A340652.
The version for factorizations is A340653.
A007716 counts non-isomorphic multiset partitions.
A007718 counts non-isomorphic connected multiset partitions.
A316980 counts non-isomorphic strict multiset partitions.
Other balance-related sequences:
- A098124 counts balanced compositions.
- A106529 lists balanced numbers.
- A340596 counts co-balanced factorizations.
- A340597 lists numbers with an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
Cf. A001055, A006141, A007717, A316983, A320663, A324522, A340654, A340655.

\\ See A340652 for G.
seq(n)={Vec(1 + sum(k=1,n,polcoef(G(n,n,k,y),k,y) - polcoef(G(n,n,k-1,y),k,y)))} \\ Andrew Howroyd, Jan 15 2024

a(11) onwards from Andrew Howroyd, Jan 15 2024

A319778 Number of non-isomorphic set systems of weight n with empty intersection whose dual is also a set system with empty intersection.

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 13, 28, 72, 181, 483

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 dual of a multiset partition has empty intersection iff no part contains all the 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) = 13 multiset partitions:
2: {{1},{2}}
3: {{1},{2},{3}}
4: {{1},{3},{2,3}}
   {{1},{2},{3},{4}}
5: {{1},{2,4},{3,4}}
   {{2},{1,3},{2,3}}
   {{1},{2},{3},{2,3}}
   {{1},{2},{4},{3,4}}
   {{1},{2},{3},{4},{5}}
6: {{3},{1,4},{2,3,4}}
   {{1,2},{1,3},{2,3}}
   {{1,3},{2,4},{3,4}}
   {{1},{2},{1,3},{2,3}}
   {{1},{2},{3,5},{4,5}}
   {{1},{3},{4},{2,3,4}}
   {{1},{3},{2,4},{3,4}}
   {{1},{4},{2,4},{3,4}}
   {{2},{3},{1,3},{2,3}}
   {{2},{4},{1,2},{3,4}}
   {{1},{2},{3},{4},{3,4}}
   {{1},{2},{3},{5},{4,5}}
   {{1},{2},{3},{4},{5},{6}}

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

Cf. A319775, A319779, A319781, A319783.

A320808 Regular tetrangle where T(n,k,i) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n, with i columns.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 0, 1, 0, 2, 4, 0, 1, 5, 4, 0, 1, 5, 5, 5, 0, 0, 1, 0, 2, 4, 0, 2, 10, 8, 0, 1, 9, 13, 7, 0, 1, 5, 12, 9, 7, 0, 0, 1, 0, 3, 6, 0, 3, 16, 12, 0, 2, 24, 33, 16, 0, 1, 14, 36, 29, 12, 0, 1, 9, 23, 29

Offset: 1

Gus Wiseman, Nov 09 2018

			Tetrangle begins:
  1  0    0      0        0          0
     0 1  0 1    0 1      0 1        0 1
          0 1 2  0 1 2    0 2 4      0 2 4
                 0 1 2 3  0 1 5 4    0 2 10 8
                          0 1 5 5 5  0 1 9 13 7
                                     0 1 5 12 9 7

Triangle sums are A007716. Triangle of row sums is A320801. Triangle of column sums is A317533. Triangle of last columns (without its leading column 1,0,0,0,...) is A055884.

Cf. A000219, A008284, A316980, A316983, A318805, A319616, A320796, A321194.

A323790 Number of non-isomorphic weight-n sets of sets of sets.

Original entry on oeis.org

1, 1, 3, 9, 33, 113, 474, 1985

Offset: 0

Gus Wiseman, Jan 27 2019

Non-isomorphic sets of sets are counted by A283877.
All sets and multisets must be finite, and only the outermost may be empty.
The weight of an atom is 1, and the weight of a multiset is the sum of weights of its elements, counting multiplicity.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 9 sets of sets of sets:
  {{1}}  {{12}}      {{123}}
         {{1}{2}}    {{1}{12}}
         {{1}}{{2}}  {{1}{23}}
                     {{1}}{{12}}
                     {{1}}{{23}}
                     {{1}{2}{3}}
                     {{1}}{{1}{2}}
                     {{1}}{{2}{3}}
                     {{1}}{{2}}{{3}}
Non-isomorphic representatives of the a(4) = 33 sets of sets of sets:
  {{1234}}             {{1}{123}}         {{1}{2}{12}}       {{1}}{{1}{12}}
  {{1}{234}}           {{12}{13}}         {{1}}{{2}{12}}
  {{12}{34}}           {{1}}{{123}}       {{12}}{{1}{2}}
  {{1}}{{234}}         {{1}{2}{13}}       {{1}}{{2}}{{12}}
  {{1}{2}{34}}         {{12}}{{13}}       {{1}}{{2}}{{1}{2}}
  {{12}}{{34}}         {{1}}{{1}{23}}
  {{1}}{{2}{34}}       {{1}}{{2}{13}}
  {{1}{2}{3}{4}}       {{12}}{{1}{3}}
  {{12}}{{3}{4}}       {{2}}{{1}{13}}
  {{1}}{{2}}{{34}}     {{1}}{{1}{2}{3}}
  {{1}}{{2}{3}{4}}     {{1}}{{2}}{{13}}
  {{1}{2}}{{3}{4}}     {{1}{2}}{{1}{3}}
  {{1}}{{2}}{{3}{4}}   {{1}}{{2}}{{1}{3}}
  {{1}}{{2}}{{3}}{{4}}

Cf. A004111, A007716, A049311, A050326, A050343, A283877, A306186, A316980, A318564, A318565, A318566, A318812.

Cf. A323787, A323788, A323789, A323791, A323792, A323793, A323794, A323795.

A319077 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 1, 3, 12, 37, 130, 428, 1481, 5091, 17979, 64176, 234311, 869645, 3295100, 12720494, 50083996, 200964437, 821845766, 3423694821, 14524845181, 62725701708, 275629610199, 1231863834775, 5597240308384, 25844969339979, 121224757935416, 577359833539428, 2791096628891679

Offset: 0

Gus Wiseman, Sep 27 2018

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.

			Non-isomorphic representatives of the a(2) = 1 through a(4) = 12 strict multiset partitions with empty intersection:
2: {{1},{2}}
3: {{1},{2,2}}
   {{1},{2,3}}
   {{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},{2},{1,2}}
   {{1},{2},{2,2}}
   {{1},{2},{3,3}}
   {{1},{2},{3,4}}
   {{1},{3},{2,3}}
   {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A318715.

Cf. A319748, A319755, A319778, A319781, A319790.

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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, k)={EulerT(Vec(sum(j=1, #q, my(g=gcd(t, q[j])); g*x^(q[j]/g)) + O(x*x^k), -k))}
R(q, n)={vector(n, t, subst(x*Ser(K(q, t, n\t)/t), x, x^t))}
a(n)={my(s=0); forpart(q=n, my(f=prod(i=1, #q, 1 - x^q[i]), u=R(q,n)); s+=permcount(q)*sum(k=0, n, my(c=polcoef(f,k)); if(c, c*polcoef(exp(sum(t=1, n\(k+1), x^(t*k)*u[t] - subst(x^(t*k)*u[t] + O(x*x^(n\2)), x, x^2), O(x*x^n) ))*if(k,1+x^k,1), n))) ); s/n!} \\ Andrew Howroyd, May 30 2023

Terms a(11) and beyond from Andrew Howroyd, May 30 2023

cp's OEIS Frontend

A323787 Number of non-isomorphic multiset partitions of strict multiset partitions of weight n.

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A319779 Number of intersecting multiset partitions of weight n whose dual is not an intersecting multiset partition.

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A319781 Number of multiset partitions of integer partitions of n with empty intersection. Number of relatively prime factorizations of Heinz numbers of integer partitions of n.

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A319755 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n.

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A319774 Number of intersecting set systems spanning n vertices whose dual is also an intersecting set system.

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A340600 Number of non-isomorphic balanced multiset partitions of weight n.

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A319778 Number of non-isomorphic set systems of weight n with empty intersection whose dual is also a set system with empty intersection.

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A320808 Regular tetrangle where T(n,k,i) is the number of nonnegative integer matrices up to row and column permutations with no zero rows or columns and k nonzero entries summing to n, with i columns.

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A323790 Number of non-isomorphic weight-n sets of sets of sets.

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A319077 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n with empty intersection.

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