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A329627 Smallest BII-number of a clutter (connected antichain) with n edges. 3
0, 1, 20, 52, 308, 820, 2868, 68404, 199476, 723764 (list; graph; refs; listen; history; text; internal format)



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.

For n > 1, a(n) appears to be the number whose binary indices are the first n terms of A018900.


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


The sequence of terms together with their corresponding set-systems begins:

0: {}

1: {{1}}

20: {{1,2},{1,3}}

52: {{1,2},{1,3},{2,3}}

308: {{1,2},{1,3},{2,3},{1,4}}

820: {{1,2},{1,3},{2,3},{1,4},{2,4}}

2868: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4}}

68404: {{1,2},{1,3},{2,3},{1,4},{2,4},{3,4},{1,5}}

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}}


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}];

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]], {2}], Length[Intersection@@s[[#]]]>0&]}, If[c=={}, s, csm[Sort[Append[Delete[s, List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];

First/@GatherBy[Select[Range[0, 10000], stableQ[bpe/@bpe[#]]&&Length[csm[bpe/@bpe[#]]]<=1&], Length[bpe[#]]&]


The version for MM-numbers is A329555.

BII-numbers of clutters are A326750.

Clutters of sets are counted by A048143.

Minimum BII-numbers of connected set-systems are A329625.

Minimum BII-numbers of antichains are A329626.

MM-numbers of connected weak antichains of multisets are A329559.

Cf. A048793, A070939, A072639, A320275, A322113, A326031, A326704, A326753, A329628, A329632.

Sequence in context: A209982 A205220 A188250 * A329628 A302885 A059677

Adjacent sequences: A329624 A329625 A329626 * A329628 A329629 A329630




Gus Wiseman, Nov 28 2019



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Last modified February 8 11:37 EST 2023. Contains 360138 sequences. (Running on oeis4.)