A326704 BII-numbers of antichains of nonempty sets.

0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 16, 18, 20, 32, 33, 36, 48, 52, 64, 128, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 256, 258, 260, 264, 266, 268, 272, 274, 276, 288, 292, 304, 308, 320, 512, 513, 516, 520, 521, 524

Offset

1,3

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every finite set of finite nonempty sets 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, it follows that the BII-number of {{2},{1,3}} is 18.

Elements of a set-system are sometimes called edges. In an antichain of sets, no edge is a subset or superset of any other edge.

Links

John Tyler Rascoe, Table of n, a(n) for n = 1..7580

John Tyler Rascoe, Python Program.

Example

The sequence of all antichains of nonempty sets together with their BII-numbers begins:
   0: {}
   1: {{1}}
   2: {{2}}
   3: {{1},{2}}
   4: {{1,2}}
   8: {{3}}
   9: {{1},{3}}
  10: {{2},{3}}
  11: {{1},{2},{3}}
  12: {{1,2},{3}}
  16: {{1,3}}
  18: {{2},{1,3}}
  20: {{1,2},{1,3}}
  32: {{2,3}}
  33: {{1},{2,3}}
  36: {{1,2},{2,3}}
  48: {{1,3},{2,3}}
  52: {{1,2},{1,3},{2,3}}

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[100], stableQ[bpe/@bpe[#], SubsetQ]&]

Prog

(Python) # see linked program

Crossrefs

Antichains of sets are counted by A000372.

Antichains of nonempty sets are counted by A014466.

MM-numbers of antichains of multisets are A316476.

BII-numbers of chains of nonempty sets are A326703.

Cf. A000120, A029931, A035327, A048793, A070939, A291166, A302521, A326031, A326675, A326701, A326702.

Sequence in context: A047453 A037467 A165564 * A309314 A309326 A326701

Adjacent sequences: A326701 A326702 A326703 * A326705 A326706 A326707

Keyword

nonn,base

Author

Gus Wiseman, Jul 21 2019

Status

approved