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A326872 BII-numbers of connectedness systems.

%I A326872 #14 Mar 07 2025 21:29:59
%S A326872 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,24,25,26,27,32,33,
%T A326872 34,35,40,41,42,43,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,
%U A326872 81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99
%N A326872 BII-numbers of connectedness systems.
%C A326872 We define a connectedness system (investigated by Vim van Dam in 2002) to be a set of finite nonempty sets (edges) that is closed under taking the union of any two overlapping edges.
%C A326872 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.
%C A326872 The enumeration of these set-systems by number of covered vertices is given by A326870.
%H A326872 John Tyler Rascoe, <a href="/A326872/b326872.txt">Table of n, a(n) for n = 1..10000</a>
%H A326872 Gus Wiseman, <a href="http://www.mathematica-journal.com/2017/12/every-clutter-is-a-tree-of-blobs/">Every Clutter Is a Tree of Blobs</a>, The Mathematica Journal, Vol. 19, 2017.
%e A326872 The sequence of all connectedness systems together with their BII-numbers begins:
%e A326872    0: {}
%e A326872    1: {{1}}
%e A326872    2: {{2}}
%e A326872    3: {{1},{2}}
%e A326872    4: {{1,2}}
%e A326872    5: {{1},{1,2}}
%e A326872    6: {{2},{1,2}}
%e A326872    7: {{1},{2},{1,2}}
%e A326872    8: {{3}}
%e A326872    9: {{1},{3}}
%e A326872   10: {{2},{3}}
%e A326872   11: {{1},{2},{3}}
%e A326872   12: {{1,2},{3}}
%e A326872   13: {{1},{1,2},{3}}
%e A326872   14: {{2},{1,2},{3}}
%e A326872   15: {{1},{2},{1,2},{3}}
%e A326872   16: {{1,3}}
%e A326872   17: {{1},{1,3}}
%e A326872   18: {{2},{1,3}}
%e A326872   19: {{1},{2},{1,3}}
%e A326872   24: {{3},{1,3}}
%e A326872   25: {{1},{3},{1,3}}
%e A326872   26: {{2},{3},{1,3}}
%e A326872   27: {{1},{2},{3},{1,3}}
%e A326872   32: {{2,3}}
%t A326872 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326872 connsysQ[eds_]:=SubsetQ[eds,Union@@@Select[Tuples[eds,2],Intersection@@#!={}&]];
%t A326872 Select[Range[0,100],connsysQ[bpe/@bpe[#]]&]
%o A326872 (Python)
%o A326872 from itertools import count, islice, combinations
%o A326872 def bin_i(n): #binary indices
%o A326872     return([(i+1) for i, x in enumerate(bin(n)[2:][::-1]) if x =='1'])
%o A326872 def a_gen():
%o A326872     for n in count(0):
%o A326872         E,f = [bin_i(k) for k in bin_i(n)],0
%o A326872         for i in combinations(E,2):
%o A326872             if list(set(i[0])|set(i[1])) not in E and len(set(i[0])&set(i[1])) > 0:
%o A326872                 f += 1
%o A326872                 break
%o A326872         if f < 1:
%o A326872             yield n
%o A326872 A326872_list = list(islice(a_gen(), 100)) # _John Tyler Rascoe_, Mar 07 2025
%Y A326872 Connectedness systems are counted by A326866, with unlabeled version A326867.
%Y A326872 The case without singletons is A326873.
%Y A326872 The connected case is A326879.
%Y A326872 Set-systems closed under union are counted by A102896, with BII numbers A326875.
%Y A326872 Cf. A029931, A048793, A072446, A326031, A326749, A326753, A326870, A326876, A326879.
%K A326872 nonn,base
%O A326872 1,3
%A A326872 _Gus Wiseman_, Jul 29 2019