cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 31-40 of 40 results.

A305856 Number of unlabeled intersecting set-systems on up to n vertices.

Original entry on oeis.org

1, 2, 4, 14, 124, 14992, 1289845584
Offset: 0

Views

Author

Gus Wiseman, Jun 11 2018

Keywords

Comments

An intersecting set-system is a finite set of finite nonempty sets (edges), any two of which have a nonempty intersection.

Examples

			Non-isomorphic representatives of the a(3) = 14 intersecting set-systems:
  {}
  {{1}}
  {{1,2}}
  {{1,2,3}}
  {{2},{1,2}}
  {{3},{1,2,3}}
  {{1,3},{2,3}}
  {{2,3},{1,2,3}}
  {{3},{1,3},{2,3}}
  {{3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3}}
  {{1,3},{2,3},{1,2,3}}
  {{3},{1,3},{2,3},{1,2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Cf. A001206, A006126, A051185, A261006, A283877, A304982, A304996, A304998, A305843, A305844, A305854-A305857.

Extensions

a(5) from Andrew Howroyd, Aug 12 2019
a(6) from Bert Dobbelaere, Apr 28 2025

A305935 Number of labeled spanning intersecting set-systems on n vertices with no singletons.

Original entry on oeis.org

1, 0, 1, 12, 809, 1146800, 899927167353, 291136684655893185321964, 14704020783497694096988185391720223222562121969, 12553242487939982849962414795232892198542733492886483991398790450208264017757788101836749760
Offset: 0

Views

Author

Gus Wiseman, Jun 15 2018

Keywords

Comments

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.

Examples

			The a(3) = 12 spanning intersecting set-systems with no singletons:
{{1,2,3}}
{{1,2},{1,3}}
{{1,2},{2,3}}
{{1,3},{2,3}}
{{1,2},{1,2,3}}
{{1,3},{1,2,3}}
{{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}}
{{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Cf. A001206, A006126, A051185, A048143, A058891, A305001, A305843, A305844, A305854-A305857, A305999, A306000, A306001, A306008.

Formula

a(n) = A305843(n) - n * A003465(n-1).
Inverse binomial transform of A306000. - Andrew Howroyd, Aug 12 2019

Extensions

a(6)-a(8) from Giovanni Resta, Jun 20 2018
a(9) from Andrew Howroyd, Aug 12 2019

A306000 Number of labeled intersecting set-systems with no singletons covering some subset of {1,...,n}.

Original entry on oeis.org

1, 1, 2, 16, 864, 1150976, 899934060544, 291136684662192699604992, 14704020783497694096990514485197495566069661696, 12553242487939982849962414795232892198542733625222671042878037323112413463887484853594095616
Offset: 0

Views

Author

Gus Wiseman, Jun 16 2018

Keywords

Comments

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

Examples

			The a(3) = 16 set-systems:
  {}
  {{1,2}}
  {{1,3}}
  {{2,3}}
  {{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,2,3}}
  {{1,3},{1,2,3}}
  {{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}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Crossrefs

Cf. A001206, A006126, A051185, A048143, A058891, A305001, A305843, A305844, A305854-A305857, A305935, A305999, A306001.

Formula

a(n) = A051185(n) - n*2^(2^(n-1)-1). - Andrew Howroyd, Aug 12 2019

Extensions

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

A326372 Number of intersecting antichains of (possibly empty) subsets of {1..n}.

Original entry on oeis.org

2, 3, 5, 13, 82, 2647, 1422565, 229809982113, 423295099074735261881
Offset: 0

Views

Author

Gus Wiseman, Jul 01 2019

Keywords

Comments

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.

Examples

			The a(0) = 2 through a(3) = 13 antichains:
  {}    {}     {}       {}
  {{}}  {{}}   {{}}     {{}}
        {{1}}  {{1}}    {{1}}
               {{2}}    {{2}}
               {{1,2}}  {{3}}
                        {{1,2}}
                        {{1,3}}
                        {{2,3}}
                        {{1,2,3}}
                        {{1,2},{1,3}}
                        {{1,2},{2,3}}
                        {{1,3},{2,3}}
                        {{1,2},{1,3},{2,3}}

Crossrefs

The case without empty edges is A001206.
The inverse binomial transform is the spanning case A305844.
The unlabeled case is A306007.
Maximal intersecting antichains are A326363.
Intersecting set systems are A051185.
Cf. A000372, A007363, A014466, A305843, A326361, A326362.

Formula

a(n) = A001206(n + 1) + 1.

A326373 Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) on n vertices.

Original entry on oeis.org

1, 1, 1, 3, 435, 989555, 887050136795, 291072121058024908202443, 14704019422368226413236661148207899662350666147, 12553242487939461785560846872353486129110194529637343578112251094358919036718815137721635299
Offset: 0

Views

Author

Gus Wiseman, Jul 01 2019

Keywords

Comments

A set system (set of sets) is intersecting if no two edges are disjoint.

Examples

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

Crossrefs

The inverse binomial transform is the covering case A326364.
Set systems with empty intersection are A318129.
Intersecting set systems are A051185.
Intersecting antichains with empty intersection are A326366.
Cf. A000371, A006126, A007363, A014466, A058891, A305844, A307249, A318128, A326361, A326362, A326363, A326365.

Programs

  • Mathematica
    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[#=={}||Intersection@@#=={}]&]],{n,0,4}]

Formula

a(n) = A051185(n) - 1 - Sum_{k=1..n-1} binomial(n,k)*A000371(k). - Andrew Howroyd, Aug 12 2019

Extensions

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

A336736 Number of factorizations of n whose distinct factors have disjoint prime signatures.

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, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 5, 1, 1, 2, 1, 1, 1
Offset: 1

Views

Author

Gus Wiseman, Aug 06 2020

Keywords

Comments

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

Examples

			The a(n) factorizations for n = 36, 360, 720, 192, 288:
  (36)     (360)    (720)     (192)      (288)
  (6*6)    (5*72)   (8*90)    (3*64)     (8*36)
  (2*2*9)  (8*45)   (9*80)    (4*48)     (9*32)
  (3*3*4)  (9*40)   (10*72)   (6*32)     (16*18)
           (10*36)  (16*45)   (12*16)    (2*144)
           (5*8*9)  (5*144)   (3*8*8)    (6*6*8)
                    (5*9*16)  (4*6*8)    (2*2*72)
                    (8*9*10)  (3*4*16)   (2*9*16)
                              (3*4*4*4)  (3*3*32)
                                         (2*2*8*9)
                                         (3*3*4*8)
                                         (2*2*2*36)
                                         (2*2*2*2*2*9)

Crossrefs

A001055 counts factorizations.
A118914 is sorted prime signature.
A124010 is prime signature.
A336737 counts factorizations with intersecting signatures.
Cf. A000372, A003182, A006126, A109298, A112798, A293606, A294068, A305844, A321469, A336424.

Programs

  • Mathematica
    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}]

A327059 Number of pairwise intersecting set-systems covering a subset of {1..n} whose dual is a weak antichain.

Original entry on oeis.org

1, 2, 4, 10, 178
Offset: 0

Views

Author

Gus Wiseman, Aug 18 2019

Keywords

Comments

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.

Examples

			The a(0) = 1 through a(3) = 10 set-systems:
  {}  {}     {}      {}
      {{1}}  {{1}}   {{1}}
             {{2}}   {{2}}
             {{12}}  {{3}}
                     {{12}}
                     {{13}}
                     {{23}}
                     {{123}}
                     {{12}{13}{23}}
                     {{12}{13}{23}{123}}

Crossrefs

Intersecting set-systems are A051185.
The BII-numbers of these set-systems are the intersection of A326910 and A326966.
Set-systems whose dual is a weak antichain are A326968.
The covering version is A327058.
The unlabeled multiset partition version is A327060.
Cf. A000372, A305844, A326951, A326965, A326970, A327057, A327062.

Programs

  • Mathematica
    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]=={}&],stableQ[dual[#],SubsetQ]&]],{n,0,3}]

Formula

Binomial transform of A327058.

A327425 Number of unlabeled antichains of nonempty sets covering n vertices where every two vertices appear together in some edge (cointersecting).

Original entry on oeis.org

1, 1, 1, 2, 6, 54
Offset: 0

Views

Author

Gus Wiseman, Sep 11 2019

Keywords

Comments

An antichain is a set of sets, none of which is a subset of any other. It is covering if there are no isolated vertices.

Examples

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 6 antichains:
    {1}  {12}  {123}         {1234}
               {12}{13}{23}  {12}{134}{234}
                             {124}{134}{234}
                             {12}{13}{14}{234}
                             {123}{124}{134}{234}
                             {12}{13}{14}{23}{24}{34}

Crossrefs

The labeled version is A327020.
Unlabeled covering antichains are A261005.
The weighted version is A327060.
Cf. A006126, A014466, A055621, A293606, A293993, A305844, A307249, A319639, A326704, A327057, A327058, A327358, A327359.

A379707 Number of nonempty labeled antichains of subsets of [n] such that all subsets except possibly those of the largest size are disjoint.

Original entry on oeis.org

1, 2, 5, 19, 133, 2605, 1128365, 68731541392, 1180735736455875189405, 170141183460507927984536600089529165335, 7237005577335553223087828975127304180898559033209149835788539833222132944557
Offset: 0

Views

Author

John Tyler Rascoe, Dec 30 2024

Keywords

Examples

			For n < 4 all nonempty labeled antichains are counted. When n=6 antichains such as {{1,2,6},{1,4},{1,5}} are not counted, while {{1,2,4},{1,2,6},{3},{5}} is counted.

Crossrefs

Cf. A000372, A014466, A229223, A245567, A305844, A379706.

Programs

  • Python
    from math import comb
def rS2(n,k,m):
    if n < 1 and k < 1: return 1
    elif n < 1 or k < 1: return 0
    else: return k*rS2(n-1,k,m) + rS2(n-1,k-1,m)- comb(n-1,m)*rS2(n-1-m,k-1,m)
def A229223(n,k):
    return sum(rS2(n,x,k) for x in range(n+1))
def A379707(n):
    return 1+sum(sum(comb(n,i)*(2**comb(n-i,s)-1)*A229223(i,s-1) for i in range(n-s+1)) for s in range(1,n+1))

Formula

a(n) = 1 + Sum_{s=1..n} (Sum_{i=0..n-s} binomial(n,i) * (2^binomial(n-i,s) - 1) * A229223(i,s-1)).

A326375 Number of intersecting antichains of subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).

Original entry on oeis.org

2, 2, 2, 3, 29, 1961, 1379274, 229755337550, 423295079757497714060
Offset: 0

Views

Author

Gus Wiseman, Jul 03 2019

Keywords

Comments

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.

Examples

			The a(4) = 29 antichains:
  {}
  {{}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{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}}

Crossrefs

The case without empty edges is A326366.
Intersecting antichains are A326372.
Antichains of nonempty sets with empty intersection are A006126 or A307249.
Cf. A001206, A007363, A014466, A051185, A058891, A305001, A305843, A305844, A318128, A318129, A326363, A326365, A326373.

Programs

  • Mathematica
    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]],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],#=={}||Intersection@@#=={}&]],{n,0,4}]

Formula

a(n) = A326366(n) + 1.

Extensions

a(7)-a(8) from Andrew Howroyd, Aug 14 2019
Previous Showing 31-40 of 40 results.

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