A319779 -id:A319779 - cp's OEIS frontend

A319752 Number of non-isomorphic intersecting multiset partitions of weight n.

1, 1, 3, 6, 16, 35, 94, 222, 584, 1488, 3977

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(4) = 16 multiset partitions:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1},{1,1,1}}
  {{1},{1,2,2}}
  {{2},{1,2,2}}
  {{3},{1,2,3}}
  {{1,1},{1,1}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,3},{2,3}}
  {{1},{1},{1,1}}
  {{2},{2},{1,2}}
  {{1},{1},{1},{1}}

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

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

A319759 Number of non-isomorphic intersecting multiset partitions of weight n with empty intersection.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 2, 13, 49, 199

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) = 13 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}}
   {{1,2},{1,3},{2,3},{2,3}}

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

Cf. A319752, A319762, A319763, A319764, A319765, A319779, A319781.

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.

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.

A319786 Number of factorizations of n where no two factors are relatively prime.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 4, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 7, 2, 2, 1, 2, 1, 4, 1, 4, 1, 1, 1, 3, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 7, 1, 1, 2, 2, 1, 1, 1, 7, 5, 1, 1, 3, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 12, 1, 2, 2, 4, 1, 1, 1, 4, 1

Offset: 1

Gus Wiseman, Sep 27 2018

nonn

First differs from A305193 at a(36) = 4, A305193(36) = 5.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 07 2018

			The a(48) = 7 factorizations are (2*2*2*6), (2*2*12), (2*4*6), (2*24), (4*12), (6*8), (48).

Antti Karttunen, Table of n, a(n) for n = 1..11520
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A007716, A045778, A049311, A162247, A281116, A283877, A305193, A305854, A306006, A316980, A318749.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319789.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],!Or@@CoprimeQ@@@Subsets[#,{2}]&]],{n,100}]

A319786(n, m=n, facs=List([])) = if(1==n, (1!=gcd(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A319786(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Nov 07 2018

More terms from Antti Karttunen, Nov 07 2018

A319787 Number of intersecting multiset partitions of normal multisets of size n.

Original entry on oeis.org

1, 1, 3, 8, 27, 95, 373, 1532, 6724

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset is normal if it spans an initial interval of positive integers.

A multiset partition is intersecting iff no two parts are disjoint.

			The a(1) = 1 through a(3) = 8 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,1,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{2},{1,2}}
   {{1},{1,2}}
   {{1},{1},{1}}

Cf. A007716, A049311, A283877, A305854, A306006, A316980, A317752.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319784, A319789.

A319760 Number of non-isomorphic intersecting strict multiset partitions (sets of multisets) of weight n.

Original entry on oeis.org

1, 1, 2, 5, 11, 26, 68, 162, 423, 1095, 2936

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(1) = 1 through a(4) = 11 strict multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
3: {{1,1,1}}
   {{1,2,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{2},{1,2}}
4: {{1,1,1,1}}
   {{1,1,2,2}}
   {{1,2,2,2}}
   {{1,2,3,3}}
   {{1,2,3,4}}
   {{1},{1,1,1}}
   {{1},{1,2,2}}
   {{2},{1,2,2}}
   {{3},{1,2,3}}
   {{1,2},{2,2}}
   {{1,3},{2,3}}

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

Cf. A319752, A319755, A319759, A319765, A319779, A319787, A319789.

A319789 Number of intersecting multiset partitions of strongly normal multisets of size n.

Original entry on oeis.org

1, 1, 3, 6, 17, 40, 122, 330, 1032

Offset: 0

Gus Wiseman, Sep 27 2018

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition its multiplicities are weakly decreasing. A multiset partition is intersecting iff no two parts are disjoint.

			The a(1) = 1 through a(3) = 6 multiset partitions:
1: {{1}}
2: {{1,1}}
   {{1,2}}
   {{1},{1}}
3: {{1,1,1}}
   {{1,1,2}}
   {{1,2,3}}
   {{1},{1,1}}
   {{1},{1,2}}
   {{1},{1},{1}}

Cf. A007716, A283877, A305854, A306006, A316980, A317755.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319784, A319787.

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.

cp's OEIS Frontend

A319752 Number of non-isomorphic intersecting multiset partitions of weight n.

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A319759 Number of non-isomorphic intersecting multiset partitions of weight n with empty intersection.

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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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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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A319786 Number of factorizations of n where no two factors are relatively prime.

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A319787 Number of intersecting multiset partitions of normal multisets of size n.

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A319760 Number of non-isomorphic intersecting strict multiset partitions (sets of multisets) of weight n.

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A319789 Number of intersecting multiset partitions of strongly normal multisets of size n.

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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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Keywords

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