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A326704 BII-numbers of antichains of nonempty sets.

%I A326704 #11 Jun 17 2024 07:05:19
%S A326704 0,1,2,3,4,8,9,10,11,12,16,18,20,32,33,36,48,52,64,128,129,130,131,
%T A326704 132,136,137,138,139,140,144,146,148,160,161,164,176,180,192,256,258,
%U A326704 260,264,266,268,272,274,276,288,292,304,308,320,512,513,516,520,521,524
%N A326704 BII-numbers of antichains of nonempty sets.
%C A326704 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.
%C A326704 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.
%H A326704 John Tyler Rascoe, <a href="/A326704/b326704.txt">Table of n, a(n) for n = 1..7580</a>
%H A326704 John Tyler Rascoe, <a href="/A326704/a326704.py.txt">Python Program</a>.
%e A326704 The sequence of all antichains of nonempty sets together with their BII-numbers begins:
%e A326704    0: {}
%e A326704    1: {{1}}
%e A326704    2: {{2}}
%e A326704    3: {{1},{2}}
%e A326704    4: {{1,2}}
%e A326704    8: {{3}}
%e A326704    9: {{1},{3}}
%e A326704   10: {{2},{3}}
%e A326704   11: {{1},{2},{3}}
%e A326704   12: {{1,2},{3}}
%e A326704   16: {{1,3}}
%e A326704   18: {{2},{1,3}}
%e A326704   20: {{1,2},{1,3}}
%e A326704   32: {{2,3}}
%e A326704   33: {{1},{2,3}}
%e A326704   36: {{1,2},{2,3}}
%e A326704   48: {{1,3},{2,3}}
%e A326704   52: {{1,2},{1,3},{2,3}}
%t A326704 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326704 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A326704 Select[Range[100],stableQ[bpe/@bpe[#],SubsetQ]&]
%o A326704 (Python) # see linked program
%Y A326704 Antichains of sets are counted by A000372.
%Y A326704 Antichains of nonempty sets are counted by A014466.
%Y A326704 MM-numbers of antichains of multisets are A316476.
%Y A326704 BII-numbers of chains of nonempty sets are A326703.
%Y A326704 Cf. A000120, A029931, A035327, A048793, A070939, A291166, A302521, A326031, A326675, A326701, A326702.
%K A326704 nonn,base
%O A326704 1,3
%A A326704 _Gus Wiseman_, Jul 21 2019