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A329626 Smallest BII-number of an antichain with n edges. 4
0, 1, 3, 11, 139, 820, 2868, 35636, 199476, 723764 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 2 06:13 EDT 2020. Contains 334767 sequences. (Running on oeis4.)