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

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

1,3

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.

Links

Table of n, a(n) for n=1..56.

Example

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}}

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]=={}&]&]

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.

Sequence in context: A185867 A326910 A326905 * A326913 A326703 A244008

Adjacent sequences: A327058 A327059 A327060 * A327062 A327063 A327064

Keyword

nonn

Author

Gus Wiseman, Aug 18 2019

Status

approved