A319787 -id:A319787 - 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.

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.

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

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.

A336737 Number of factorizations of n whose factors have pairwise intersecting prime signatures.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 7, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 6, 3, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

First differs from A327400 at a(72) = 9, A327400(72) = 10.

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization.

			The a(n) factorizations for n = 2, 4, 12, 24, 30, 36, 60:
  (2)  (4)    (12)     (24)       (30)     (36)       (60)
       (2*2)  (2*6)    (2*12)     (5*6)    (4*9)      (2*30)
              (2*2*3)  (2*2*6)    (2*15)   (6*6)      (3*20)
                       (2*2*2*3)  (3*10)   (2*18)     (5*12)
                                  (2*3*5)  (3*12)     (6*10)
                                           (2*3*6)    (2*5*6)
                                           (2*2*3*3)  (2*2*15)
                                                      (2*3*10)
                                                      (2*2*3*5)

A001055 counts factorizations.
A118914 is sorted prime signature.
A124010 is prime signature.
A336736 counts factorizations with disjoint signatures.
Cf. A003182, A051185, A305843, A305844, A305854, A306006, A319752, A319787, A319789, A321469, A336424.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
Table[Length[Select[facs[n],stableQ[#,Intersection[prisig[#1],prisig[#2]]=={}&]&]],{n,100}]

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.

A319784 Number of non-isomorphic intersecting T_0 set systems of weight n.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 3, 5, 7, 14, 25

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. 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(8) = 7 multiset partitions:
1: {{1}}
3: {{2},{1,2}}
4: {{1,3},{2,3}}
5: {{3},{1,3},{2,3}}
6: {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,4},{2,4},{3,4}}
7: {{4},{1,3,4},{2,3,4}}
   {{1,3},{1,4},{2,3,4}}
   {{1,3},{2,3},{1,2,3}}
   {{1,4},{3,4},{2,3,4}}
   {{4},{1,4},{2,4},{3,4}}
8: {{1,5},{2,4,5},{3,4,5}}
   {{2,4},{3,4},{1,2,3,4}}
   {{2,4},{1,2,5},{3,4,5}}
   {{2,4},{1,3,4},{2,3,4}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{4},{1,4},{3,4},{2,3,4}}
   {{1,5},{2,5},{3,5},{4,5}}

Cf. A007716, A049311, A283877, A305843, A305854, A306006, A316980, A317752.

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

cp's OEIS Frontend

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

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

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

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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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A336737 Number of factorizations of n whose factors have pairwise intersecting prime signatures.

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

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A319784 Number of non-isomorphic intersecting T_0 set systems of weight n.

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