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A326947 BII-numbers of T_0 set-systems.

%I A326947 #9 Jul 26 2024 05:28:26
%S A326947 0,1,2,3,5,6,7,8,9,10,11,13,14,15,17,19,20,21,22,23,24,25,26,27,28,29,
%T A326947 30,31,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,
%U A326947 55,56,57,58,59,60,61,62,63,67,69,70,71,73,74,75,77,78
%N A326947 BII-numbers of T_0 set-systems.
%C A326947 The dual of a set-system has, for each vertex, one block consisting of the indices (or positions) of the blocks containing that vertex. For example, the dual of {{1,2},{2,3}} is {{1},{1,2},{2}}. The T_0 condition means that the dual is strict (no repeated edges).
%C A326947 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 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 A326947 John Tyler Rascoe, <a href="/A326947/b326947.txt">Table of n, a(n) for n = 1..10000</a>
%e A326947 The sequence of all T_0 set-systems together with their BII numbers begins:
%e A326947    0: {}
%e A326947    1: {{1}}
%e A326947    2: {{2}}
%e A326947    3: {{1},{2}}
%e A326947    5: {{1},{1,2}}
%e A326947    6: {{2},{1,2}}
%e A326947    7: {{1},{2},{1,2}}
%e A326947    8: {{3}}
%e A326947    9: {{1},{3}}
%e A326947   10: {{2},{3}}
%e A326947   11: {{1},{2},{3}}
%e A326947   13: {{1},{1,2},{3}}
%e A326947   14: {{2},{1,2},{3}}
%e A326947   15: {{1},{2},{1,2},{3}}
%e A326947   17: {{1},{1,3}}
%e A326947   19: {{1},{2},{1,3}}
%e A326947   20: {{1,2},{1,3}}
%e A326947   21: {{1},{1,2},{1,3}}
%e A326947   22: {{2},{1,2},{1,3}}
%e A326947   23: {{1},{2},{1,2},{1,3}}
%t A326947 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326947 dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
%t A326947 TZQ[sys_]:=UnsameQ@@dual[sys];
%t A326947 Select[Range[0,100],TZQ[bpe/@bpe[#]]&]
%o A326947 (Python)
%o A326947 from itertools import count, chain, islice
%o A326947 def bin_i(n): #binary indices
%o A326947     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o A326947 def a_gen():
%o A326947     for n in count(0):
%o A326947         a,b,s = [bin_i(k) for k in bin_i(n)],[],set()
%o A326947         for i in {i for i in chain.from_iterable(a)}:
%o A326947             b.append([])
%o A326947             for j in range(len(a)):
%o A326947                 if i in a[j]:
%o A326947                     b[-1].append(j)
%o A326947             s.add(tuple(b[-1]))
%o A326947         if len(s) == len(b):
%o A326947             yield n
%o A326947 A326947_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, Jul 25 2024
%Y A326947 T_0 set-systems are counted by A326940, with unlabeled version A326946.
%Y A326947 Cf. A059201, A316978, A319559, A319564, A326939, A326941, A326949.
%K A326947 nonn,base
%O A326947 1,3
%A A326947 _Gus Wiseman_, Aug 08 2019