A329626 Smallest BII-number of an antichain with n edges.

0, 1, 3, 11, 139, 820, 2868, 35636, 199476, 723764

Offset

0,3

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.

Links

Table of n, a(n) for n=0..9.

Example

The sequence of terms together with their corresponding set-systems begins:
       0: {}
       1: {{1}}
       3: {{1},{2}}
      11: {{1},{2},{3}}
     139: {{1},{2},{3},{4}}
     820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
    2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}
   35636: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{5}}
  199476: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5}}
  723764: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5},{2,5},{3,5}}

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_]:=!Apply[Or, Outer[#1=!=#2&&SubsetQ[#1, #2]&, u, u, 1], {0, 1}];
First/@GatherBy[Select[Range[0, 10000], stableQ[bpe/@bpe[#]]&], Length[bpe[#]]&]

Crossrefs

The connected case is A329627.

The intersecting case is A329628.

BII-numbers of antichains are A326704.

Antichain covers are A006126.

Cf. A048143, A048793, A070939, A303362, A319837, A326031, A326750, A329555, A329560, A329561, A329625.

Sequence in context: A121897 A067657 A063502 * A072639 A201000 A100459

Adjacent sequences: A329623 A329624 A329625 * A329627 A329628 A329629

Keyword

nonn,more

Author

Gus Wiseman, Nov 28 2019

Status

approved