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A326749 BII-numbers of connected set-systems.

%I A326749 #15 Jul 26 2024 07:59:31
%S A326749 0,1,2,4,5,6,7,8,16,17,20,21,22,23,24,25,28,29,30,31,32,34,36,37,38,
%T A326749 39,40,42,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,
%U A326749 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82
%N A326749 BII-numbers of connected set-systems.
%C A326749 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, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
%H A326749 John Tyler Rascoe, <a href="/A326749/b326749.txt">Table of n, a(n) for n = 1..10000</a>
%e A326749 The sequence of all connected set-systems together with their BII-numbers begins:
%e A326749    0: {}
%e A326749    1: {{1}}
%e A326749    2: {{2}}
%e A326749    4: {{1,2}}
%e A326749    5: {{1},{1,2}}
%e A326749    6: {{2},{1,2}}
%e A326749    7: {{1},{2},{1,2}}
%e A326749    8: {{3}}
%e A326749   16: {{1,3}}
%e A326749   17: {{1},{1,3}}
%e A326749   20: {{1,2},{1,3}}
%e A326749   21: {{1},{1,2},{1,3}}
%e A326749   22: {{2},{1,2},{1,3}}
%e A326749   23: {{1},{2},{1,2},{1,3}}
%e A326749   24: {{3},{1,3}}
%e A326749   25: {{1},{3},{1,3}}
%e A326749   28: {{1,2},{3},{1,3}}
%e A326749   29: {{1},{1,2},{3},{1,3}}
%e A326749   30: {{2},{1,2},{3},{1,3}}
%e A326749   31: {{1},{2},{1,2},{3},{1,3}}
%t A326749 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326749 csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
%t A326749 Select[Range[0,100],Length[csm[bpe/@bpe[#]]]<=1&]
%o A326749 (Python)
%o A326749 from itertools import count, islice
%o A326749 from sympy.utilities.iterables import connected_components
%o A326749 def bin_i(n): #binary indices
%o A326749     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o A326749 def a_gen():
%o A326749     yield 0
%o A326749     for n in count(1):
%o A326749         a, E = [bin_i(k) for k in bin_i(n)], []
%o A326749         m = len(a)
%o A326749         for i in range(m):
%o A326749             for j in a[i]:
%o A326749                 for k in range(m):
%o A326749                     if j in a[k]:
%o A326749                         E.append((i, k))
%o A326749         for v in connected_components((list(range(m)), E)):
%o A326749             if len(v) == m:
%o A326749                 yield n
%o A326749 A326749_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, Jul 25 2024
%Y A326749 Positions of 0's and 1's in A326753.
%Y A326749 Cf. A000120, A001187, A029931, A048143, A048793, A070939, A072639, A323818, A326031, A326702.
%Y A326749 Other BII-numbers: A309314 (hyperforests), A326701 (set partitions), A326703 (chains), A326704 (antichains), A326750 (clutters), A326751 (blobs), A326752 (hypertrees), A326754 (covers).
%K A326749 nonn,base
%O A326749 1,3
%A A326749 _Gus Wiseman_, Jul 23 2019