A329560 BII-numbers of antichains of sets with empty intersection.

0, 3, 9, 10, 11, 12, 18, 33, 52, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 258, 264, 266, 268, 274, 288, 292, 304, 308, 513, 520, 521, 524, 528, 532, 545, 560, 564, 772, 776, 780, 784, 788, 800, 804, 816, 820, 832

Offset

1,2

Comments

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 (finite set of finite nonempty sets of positive integers) 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. Elements of a set-system are sometimes called edges.

A set-system is an antichain if no edge is a proper subset of any other.

Empty intersection means there is no vertex in common to all the edges

Links

Table of n, a(n) for n=1..55.

Example

The sequence of terms together with their binary expansions and corresponding set-systems begins:
    0:          0 ~ {}
    3:         11 ~ {{1},{2}}
    9:       1001 ~ {{1},{3}}
   10:       1010 ~ {{2},{3}}
   11:       1011 ~ {{1},{2},{3}}
   12:       1100 ~ {{1,2},{3}}
   18:      10010 ~ {{2},{1,3}}
   33:     100001 ~ {{1},{2,3}}
   52:     110100 ~ {{1,2},{1,3},{2,3}}
  129:   10000001 ~ {{1},{4}}
  130:   10000010 ~ {{2},{4}}
  131:   10000011 ~ {{1},{2},{4}}
  132:   10000100 ~ {{1,2},{4}}
  136:   10001000 ~ {{3},{4}}
  137:   10001001 ~ {{1},{3},{4}}
  138:   10001010 ~ {{2},{3},{4}}
  139:   10001011 ~ {{2},{3},{4}}
  140:   10001100 ~ {{1,2},{3},{4}}
  144:   10010000 ~ {{1,3},{4}}
  146:   10010010 ~ {{2},{1,3},{4}}
  148:   10010100 ~ {{1,2},{1,3},{4}}

Mathematica

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], #==0||Intersection@@bpe/@bpe[#]=={}&&stableQ[bpe/@bpe[#], SubsetQ]&]

Crossrefs

Intersection of A326911 and A326704.

BII-numbers of intersecting set-systems with empty intersecting are A326912.

Cf. A000120, A048793, A070939, A326031, A326701, A328671, A329561, A329626, A329628, A329661.

Sequence in context: A228165 A114022 A181862 * A330296 A043048 A308421

Adjacent sequences: A329557 A329558 A329559 * A329561 A329562 A329563

Keyword

nonn

Author

Gus Wiseman, Nov 28 2019

Status

approved