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.

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

Original entry on oeis.org

1, 1, 3, 62, 24710, 2076948136, 9221293198653529144, 170141182628636920684331812494864430896
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}}. This sequence counts covering set-systems whose dual is strict and pairwise intersecting.

Examples

			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

Crossrefs

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.

Programs

  • Mathematica
    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}]

Formula

Inverse binomial transform of A327052.

Extensions

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

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

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}}. 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.

Examples

			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}}

Crossrefs

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.

Programs

  • Mathematica
    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]=={}&]&]

A327061 BII-numbers of pairwise intersecting set-systems where every two covered vertices appear together in some edge (cointersecting).

Original entry on oeis.org

0, 1, 2, 4, 5, 6, 8, 16, 17, 24, 32, 34, 40, 52, 64, 65, 66, 68, 69, 70, 72, 80, 81, 84, 85, 88, 96, 98, 100, 102, 104, 112, 116, 120, 128, 256, 257, 384, 512, 514, 640, 772, 1024, 1025, 1026, 1028, 1029, 1030, 1152, 1280, 1281, 1284, 1285, 1408, 1536, 1538
Offset: 1

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}}. This sequence gives all BII-numbers (defined below) of pairwise intersecting set-systems whose dual is also 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.

Examples

			The sequence of all pairwise intersecting, cointersecting set-systems together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   4: {{1,2}}
   5: {{1},{1,2}}
   6: {{2},{1,2}}
   8: {{3}}
  16: {{1,3}}
  17: {{1},{1,3}}
  24: {{3},{1,3}}
  32: {{2,3}}
  34: {{2},{2,3}}
  40: {{3},{2,3}}
  52: {{1,2},{1,3},{2,3}}
  64: {{1,2,3}}
  65: {{1},{1,2,3}}
  66: {{2},{1,2,3}}
  68: {{1,2},{1,2,3}}
  69: {{1},{1,2},{1,2,3}}
  70: {{2},{1,2},{1,2,3}}

Crossrefs

The unlabeled multiset partition version is A319765.
Equals the intersection of A326853 and A326910.
The T_0 version is A326854.
These set-systems are counted by A327037 (covering) and A327038 (not covering).
Cf. A029931, A048793, A051185, A305843, A319774, A326031, A327039, A327040, A327041, A327057.

Programs

  • Mathematica
    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[0,100],stableQ[bpe/@bpe[#],Intersection[#1,#2]=={}&]&&stableQ[dual[bpe/@bpe[#]],Intersection[#1,#2]=={}&]&]
Previous Showing 11-13 of 13 results.

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