A305843 -id:A305843 - cp's OEIS frontend

A327053 Number of T_0 (costrict) set-systems covering n vertices where every two vertices appear together in some edge (cointersecting).

1, 1, 3, 62, 24710, 2076948136, 9221293198653529144, 170141182628636920684331812494864430896

Offset: 0

Gus Wiseman, Aug 18 2019

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. This sequence counts covering set-systems whose dual is strict and pairwise intersecting.

			The a(1) = 1 through a(2) = 3 set-systems:
  {}  {{1}}  {{1},{1,2}}
             {{2},{1,2}}
             {{1},{2},{1,2}}
The a(3) = 62 set-systems:
  1 2 123    1 2 3 123    1 2 12 13 23   1 2 3 12 13 23   1 2 3 12 13 23 123
  1 3 123    1 12 13 23   1 2 3 12 123   1 2 3 12 13 123
  2 3 123    1 2 12 123   1 2 3 13 123   1 2 3 12 23 123
  1 12 123   1 2 13 123   1 2 3 23 123   1 2 3 13 23 123
  1 13 123   1 2 23 123   1 3 12 13 23   1 2 12 13 23 123
  12 13 23   1 3 12 123   2 3 12 13 23   1 3 12 13 23 123
  2 12 123   1 3 13 123   1 2 12 13 123  2 3 12 13 23 123
  2 23 123   1 3 23 123   1 2 12 23 123
  3 13 123   2 12 13 23   1 2 13 23 123
  3 23 123   2 3 12 123   1 3 12 13 123
  12 13 123  2 3 13 123   1 3 12 23 123
  12 23 123  2 3 23 123   1 3 13 23 123
  13 23 123  3 12 13 23   2 3 12 13 123
             1 12 13 123  2 3 12 23 123
             1 12 23 123  2 3 13 23 123
             1 13 23 123  1 12 13 23 123
             2 12 13 123  2 12 13 23 123
             2 12 23 123  3 12 13 23 123
             2 13 23 123
             3 12 13 123
             3 12 23 123
             3 13 23 123
             12 13 23 123

The pairwise intersecting case is A319774.
The BII-numbers of these set-systems are the intersection of A326947 and A326853.
The non-T_0 version is A327040.
The non-covering version is A327052.
Cf. A003465, A305843, A319767, A326854, A327020, A327037, A327039.

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[dual[#],Intersection[#1,#2]=={}&]&]],{n,0,3}]

Inverse binomial transform of A327052.

a(5)-a(7) from Christian Sievers, Feb 04 2024

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

A326365 Number of intersecting antichains with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

Original entry on oeis.org

1, 0, 0, 1, 23, 1834, 1367903, 229745722873, 423295077919493525420

Offset: 0

Gus Wiseman, Jul 01 2019

Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 23 intersecting antichains with empty intersection:
  {{1,2},{1,3},{2,3,4}}
  {{1,2},{1,4},{2,3,4}}
  {{1,2},{2,3},{1,3,4}}
  {{1,2},{2,4},{1,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,3},{1,2,4}}
  {{1,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{1,2,3}}
  {{1,4},{3,4},{1,2,3}}
  {{2,3},{2,4},{1,3,4}}
  {{2,3},{3,4},{1,2,4}}
  {{2,4},{3,4},{1,2,3}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Intersecting antichain covers are A305844.
Intersecting covers with empty intersection are A326364.
Antichain covers with empty intersection are A305001.
The binomial transform is the non-covering case A326366.
Covering, intersecting antichains with empty intersection are A326365.
Cf. A006126, A007363, A014466, A051185, A058891, A305843, A307249, A318128, A318129, A326361, A326362, A326363.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,4}]

a(7)-a(8) from Andrew Howroyd, Aug 14 2019

A326366 Number of intersecting antichains of nonempty subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).

Original entry on oeis.org

1, 1, 1, 2, 28, 1960, 1379273, 229755337549, 423295079757497714059

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no edge is a subset of any other, and is intersecting if no two edges are disjoint.

			The a(0) = 1 through a(4) = 28 intersecting antichains with empty intersection:
  {}  {}  {}  {}              {}
              {{12}{13}{23}}  {{12}{13}{23}}
                              {{12}{14}{24}}
                              {{13}{14}{34}}
                              {{23}{24}{34}}
                              {{12}{13}{234}}
                              {{12}{14}{234}}
                              {{12}{23}{134}}
                              {{12}{24}{134}}
                              {{13}{14}{234}}
                              {{13}{23}{124}}
                              {{13}{34}{124}}
                              {{14}{24}{123}}
                              {{14}{34}{123}}
                              {{23}{24}{134}}
                              {{23}{34}{124}}
                              {{24}{34}{123}}
                              {{12}{134}{234}}
                              {{13}{124}{234}}
                              {{14}{123}{234}}
                              {{23}{124}{134}}
                              {{24}{123}{134}}
                              {{34}{123}{124}}
                              {{12}{13}{14}{234}}
                              {{12}{23}{24}{134}}
                              {{13}{23}{34}{124}}
                              {{14}{24}{34}{123}}
                              {{123}{124}{134}{234}}

The case with empty edges allowed is A326375.
Intersecting antichains of nonempty sets are A001206.
Intersecting set systems with empty intersection are A326373.
Antichains of nonempty sets with empty intersection are A006126 or A307249.
The inverse binomial transform is the covering case A326365.
Cf. A007363, A014466, A051185, A058891, A305001, A305843, A305844, A318128, A318129, A326361, A326362, A326363, A326364.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],#=={}||Intersection@@#=={}&]],{n,0,4}]

a(n) = A326375(n) - 1.

a(n) = A001206(n+1) + A307249(n) - A014466(n). - Andrew Howroyd, Aug 14 2019

a(7)-a(8) from Andrew Howroyd, Aug 14 2019

A326854 BII-numbers of T_0 (costrict), pairwise intersecting set-systems where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

0, 1, 2, 5, 6, 8, 17, 24, 34, 40, 52, 69, 70, 81, 84, 85, 88, 98, 100, 102, 104, 112, 116, 120, 128, 257, 384, 514, 640, 772, 1029, 1030, 1281, 1284, 1285, 1408, 1538, 1540, 1542, 1664, 1792, 1796, 1920, 2056, 2176, 2320, 2592, 2880, 3120, 3152, 3168, 3184

Offset: 1

Gus Wiseman, Aug 18 2019

nonn

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. This sequence gives all BII-numbers (defined below) of pairwise intersecting set-systems whose dual is strict and pairwise intersecting.

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

			The sequence of all set-systems that are pairwise intersecting, cointersecting, and costrict, together with their BII-numbers, begins:
    0: {}
    1: {{1}}
    2: {{2}}
    5: {{1},{1,2}}
    6: {{2},{1,2}}
    8: {{3}}
   17: {{1},{1,3}}
   24: {{3},{1,3}}
   34: {{2},{2,3}}
   40: {{3},{2,3}}
   52: {{1,2},{1,3},{2,3}}
   69: {{1},{1,2},{1,2,3}}
   70: {{2},{1,2},{1,2,3}}
   81: {{1},{1,3},{1,2,3}}
   84: {{1,2},{1,3},{1,2,3}}
   85: {{1},{1,2},{1,3},{1,2,3}}
   88: {{3},{1,3},{1,2,3}}
   98: {{2},{2,3},{1,2,3}}
  100: {{1,2},{2,3},{1,2,3}}
  102: {{2},{1,2},{2,3},{1,2,3}}

Equals the intersection of A326947, A326910, and A326853.
These set-systems are counted by A319774 (covering).
The non-T_0 version is A327061.
Cf. A029931, A048793, A051185, A305843, A319765, A326031, A327037, A327038, A327041, A327052, A327053.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,10000],UnsameQ@@dual[bpe/@bpe[#]]&&stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&&stableQ[dual[bpe/@bpe[#]],Intersection[#1,#2]=={}&]&]

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.

A326364 Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

Original entry on oeis.org

1, 0, 0, 2, 426, 987404, 887044205940, 291072121051815578010398, 14704019422368226413234332571239460300433492086, 12553242487939461785560846872353486129110194397301168776798213375239447299205732561174066488

Offset: 0

Gus Wiseman, Jul 01 2019

nonn

Covering means there are no isolated vertices. A set system (set of sets) is intersecting if no two edges are disjoint.

			The a(3) = 2 intersecting set systems with empty intersection:
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Covering set systems with empty intersection are A318128.
Covering, intersecting set systems are A305843.
Covering, intersecting antichains with empty intersection are A326365.
Cf. A006126, A007363, A014466, A051185, A058891, A305844, A307249, A318129, A326361, A326362, A326363.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,4}]

Inverse binomial transform of A326373. - Andrew Howroyd, Aug 12 2019

a(6)-a(9) from Andrew Howroyd, Aug 12 2019

A327058 Number of pairwise intersecting set-systems covering n vertices whose dual is a weak antichain.

Original entry on oeis.org

1, 1, 1, 3, 155

Offset: 0

Gus Wiseman, Aug 18 2019

A set-system is a finite set of finite nonempty sets. Its elements are sometimes called edges. The dual of a set-system has, for each vertex, one edge consisting of the indices (or positions) of the edges containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. A weak antichain is a multiset of sets, none of which is a proper subset of any other.

			The a(0) = 1 through a(3) = 3 set-systems:
  {}  {{1}}  {{12}}  {{123}}
                     {{12}{13}{23}}
                     {{12}{13}{23}{123}}

Covering intersecting set-systems are A305843.
The BII-numbers of these set-systems are the intersection of A326910 and A326966.
Covering coantichains are A326970.
The non-covering version is A327059.
The unlabeled multiset partition version is A327060.
Cf. A006126, A051185, A059523, A305844, A319639, A326961, A326965, A326968, A327020, A327057.

dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],Union@@#==Range[n]&&stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Inverse binomial transform of A327059.

A337983 Number of compositions of n into distinct parts, any two of which have a common divisor > 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 3, 1, 3, 3, 5, 1, 13, 1, 13, 7, 19, 1, 35, 1, 59, 15, 65, 1, 117, 5, 133, 27, 195, 1, 411, 7, 435, 67, 617, 17, 941, 7, 1177, 135, 1571, 13, 2939, 31, 3299, 375, 4757, 13, 6709, 43, 8813, 643, 11307, 61, 16427, 123, 24331, 1203, 30461, 67

Offset: 0

Gus Wiseman, Oct 06 2020

nonn

Number of pairwise non-coprime strict compositions of n.

			The a(2) = 1 through a(15) = 7 compositions (A..F = 10..15):
  2  3  4  5  6   7  8   9   A   B  C    D  E    F
              24     26  36  28     2A      2C   3C
              42     62  63  46     39      4A   5A
                             64     48      68   69
                             82     84      86   96
                                    93      A4   A5
                                    A2      C2   C3
                                    246     248
                                    264     284
                                    426     428
                                    462     482
                                    624     824
                                    642     842

A318717 is the unordered version.
A318719 is the version for Heinz numbers of partitions.
A337561 is the pairwise coprime instead of pairwise non-coprime version, or A337562 if singletons are considered coprime.
A337605*6 counts these compositions of length 3.
A337667 is the non-strict version, ranked by A337666.
A337696 ranks these compositions.
A051185 and A305843 (covering) count pairwise intersecting set-systems.
A101268 counts pairwise coprime or singleton compositions.
A200976 and A328673 are the unordered version.
A233564 ranks strict compositions.
A318749 is the version for factorizations, with non-strict version A319786.
A333228 ranks compositions whose distinct parts are pairwise coprime.
A335236 ranks compositions neither a singleton nor pairwise coprime.
A337462 counts pairwise coprime compositions.
A337694 lists numbers with no two relatively prime prime indices.
Cf. A082024, A284825, A302797, A305713, A319752, A327516, A333227, A335235, A336737, A337604.

stabQ[u_,Q_]:=And@@Not/@Q@@@Tuples[u,2];
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&&stabQ[#,CoprimeQ]&]],{n,0,30}]

cp's OEIS Frontend

A327053 Number of T_0 (costrict) set-systems covering n vertices where every two vertices appear together in some edge (cointersecting).

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

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A326365 Number of intersecting antichains with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

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A326366 Number of intersecting antichains of nonempty subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).

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A326854 BII-numbers of T_0 (costrict), pairwise intersecting set-systems where every two vertices appear together in some edge (cointersecting).

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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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A326364 Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

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