A305854 -id:A305854 - cp's OEIS frontend

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

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.

A305999 Number of unlabeled spanning intersecting set-systems on n vertices with no singletons.

Original entry on oeis.org

1, 0, 1, 6, 76, 12916

Offset: 0

Gus Wiseman, Jun 16 2018

An intersecting set-system S is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. S is spanning if every vertex is contained in some edge. A singleton is an edge containing only one vertex.

			Non-isomorphic representative of the a(3) = 6 set-systems:
{{1,2,3}}
{{1,3},{2,3}}
{{2,3},{1,2,3}}
{{1,2},{1,3},{2,3}}
{{1,3},{2,3},{1,2,3}}
{{1,2},{1,3},{2,3},{1,2,3}}

Cf. A001206, A051185, A048143, A261006, A058891, A261005, A304998, A305854-A305857, A305935, A306000, A306001, A306008.

a(n) = A306001(n) - A306001(n-1) for n > 0. - Andrew Howroyd, Aug 12 2019

a(5) from Andrew Howroyd, Aug 12 2019

A306008 Number of non-isomorphic intersecting set-systems of weight n with no singletons.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 7, 10, 21, 39, 78

Offset: 0

Gus Wiseman, Jun 16 2018

An intersecting set-system is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection. The weight of a set-system is the sum of cardinalities of its elements. Weight is generally not the same as number of vertices.

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

Cf. A007716, A034691, A048143, A049311, A116540, A283877, A293606, A293607, A304867, A305999, A305854-A305857, A306005-A306007.

A319766 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n whose dual is also a strict intersecting multiset partition.

Original entry on oeis.org

1, 1, 1, 4, 6, 14, 31, 64, 145, 324, 753

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 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(5) = 14 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}}
5: {{1,1,1,1,1}}
   {{1,1,2,2,2}}
   {{1,2,2,2,2}}
   {{1},{1,1,1,1}}
   {{1},{1,2,2,2}}
   {{2},{1,1,2,2}}
   {{2},{1,2,2,2}}
   {{2},{1,2,3,3}}
   {{1,1},{1,1,1}}
   {{1,1},{1,2,2}}
   {{1,2},{1,2,2}}
   {{1,2},{2,2,2}}
   {{2,2},{1,2,2}}
   {{2},{1,2},{2,2}}

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

Cf. A319752, A319765, A319767, A319768, A319769, A319773, A319774.

A319768 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n whose dual is a (not necessarily strict) intersecting multiset partition.

Original entry on oeis.org

1, 1, 2, 5, 11, 25, 63, 144, 364, 905, 2356

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(1) = 1 through a(4) = 11 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},{2},{1,2}}

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

Cf. A319752, A319765, A319766, A319767, A319769, A319773, A319774.

A319769 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n whose dual is also an intersecting set multipartition.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 12, 16, 26, 38, 61

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(1) = 1 through a(5) = 7 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}}
   {{2},{2},{1,2}}
   {{1},{1},{1},{1}}
5: {{1,2,3,4,5}}
   {{4},{1,2,3,4}}
   {{2,3},{1,2,3}}
   {{2},{1,2},{1,2}}
   {{3},{3},{1,2,3}}
   {{2},{2},{2},{1,2}}
   {{1},{1},{1},{1},{1}}

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

Cf. A319752, A319765, A319766, A319767, A319768, A319773, A319774.

A319773 Number of non-isomorphic intersecting set systems of weight n whose dual is also an intersecting set system.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 2, 1, 2, 4, 5

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(1) = 1 through a(10) = 5 set systems:
1:  {{1}}
3:  {{2},{1,2}}
6:  {{3},{2,3},{1,2,3}}
    {{1,2},{1,3},{2,3}}
7:  {{1,3},{2,3},{1,2,3}}
8:  {{2,4},{3,4},{1,2,3,4}}
    {{3},{1,3},{2,3},{1,2,3}}
9:  {{1,2,4},{1,3,4},{2,3,4}}
    {{4},{2,4},{3,4},{1,2,3,4}}
    {{1,2},{1,3},{1,4},{2,3,4}}
    {{1,2},{1,3},{2,3},{1,2,3}}
10: {{4},{3,4},{2,3,4},{1,2,3,4}}
    {{4},{1,2,4},{1,3,4},{2,3,4}}
    {{1,2},{2,4},{1,3,4},{2,3,4}}
    {{1,4},{2,4},{3,4},{1,2,3,4}}
    {{2,3},{2,4},{3,4},{1,2,3,4}}

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

Cf. A319752, A319765, A319766, A319767, A319768, A319769, A319774.

A318749 Number of pairwise relatively nonprime strict factorizations of n (no two factors are coprime).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Oct 08 2018

			The a(96) = 7 factorizations are (96), (2*48), (4*24), (6*16), (8*12), (2*4*12), (2*6*8).
The a(480) = 18 factorizations:
  (480)
  (2*240) (4*120) (6*80) (8*60) (10*48) (12*40) (16*30) (20*24)
  (2*4*60) (2*6*40) (2*8*30) (2*10*24) (2*12*20) (4*6*20) (4*10*12) (6*8*10)
  (2*4*6*10)

Antti Karttunen, Table of n, a(n) for n = 1..10000
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, A045778, A051185, A281116, A303282, A305843, A305854, A317748, A318715, A318717, A318721.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[strfacs[n/d],Min@@#1>d&],{d,Rest[Divisors[n]]}]];
Table[Length[Select[strfacs[n],And@@(GCD[##]>1&)@@@Select[Tuples[#,2],Less@@#&]&]],{n,50}]

A318749(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 += A318749(n/d, d-1, newfacs))); (s)); \\ Antti Karttunen, Oct 08 2018

More terms from Antti Karttunen, Oct 08 2018

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}]

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

A305999 Number of unlabeled spanning intersecting set-systems on n vertices with no singletons.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions

A306008 Number of non-isomorphic intersecting set-systems of weight n with no singletons.

Views

Author

Keywords

Comments

Examples

Crossrefs

A319766 Number of non-isomorphic strict intersecting multiset partitions (sets of multisets) of weight n whose dual is also a strict intersecting multiset partition.

Views

Author

Keywords

Comments

Examples

Crossrefs

A319768 Number of non-isomorphic strict multiset partitions (sets of multisets) of weight n whose dual is a (not necessarily strict) intersecting multiset partition.

Views

Author

Keywords

Comments

Examples

Crossrefs

A319769 Number of non-isomorphic intersecting set multipartitions (multisets of sets) of weight n whose dual is also an intersecting set multipartition.

Views

Author

Keywords

Comments

Examples

Crossrefs

A319773 Number of non-isomorphic intersecting set systems of weight n whose dual is also an intersecting set system.

Views

Author

Keywords

Comments

Examples

Crossrefs

A318749 Number of pairwise relatively nonprime strict factorizations of n (no two factors are coprime).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica