cp's OEIS Frontend

A326879 BII-numbers of connected connectedness systems.

%I A326879 #7 Jul 31 2019 09:59:57
%S A326879 0,1,2,4,5,6,7,8,16,17,24,25,32,34,40,42,64,65,66,67,68,69,70,71,72,
%T A326879 73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,
%U A326879 96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112
%N A326879 BII-numbers of connected connectedness systems.
%C A326879 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. It is connected if it contains an edge containing all the vertices.
%C A326879 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 A326879 The enumeration of connected connectedness systems by number of vertices is given by A326868.
%H A326879 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 A326879 The sequence of all connected connectedness systems together with their BII-numbers begins:
%e A326879    0: {}
%e A326879    1: {{1}}
%e A326879    2: {{2}}
%e A326879    4: {{1,2}}
%e A326879    5: {{1},{1,2}}
%e A326879    6: {{2},{1,2}}
%e A326879    7: {{1},{2},{1,2}}
%e A326879    8: {{3}}
%e A326879   16: {{1,3}}
%e A326879   17: {{1},{1,3}}
%e A326879   24: {{3},{1,3}}
%e A326879   25: {{1},{3},{1,3}}
%e A326879   32: {{2,3}}
%e A326879   34: {{2},{2,3}}
%e A326879   40: {{3},{2,3}}
%e A326879   42: {{2},{3},{2,3}}
%e A326879   64: {{1,2,3}}
%e A326879   65: {{1},{1,2,3}}
%e A326879   66: {{2},{1,2,3}}
%e A326879   67: {{1},{2},{1,2,3}}
%e A326879   68: {{1,2},{1,2,3}}
%t A326879 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A326879 connsysQ[eds_]:=SubsetQ[eds,Union@@@Select[Tuples[eds,2],Intersection@@#!={}&]];
%t A326879 Select[Range[0,100],#==0||MemberQ[bpe/@bpe[#],Union@@bpe/@bpe[#]]&&connsysQ[bpe/@bpe[#]]&]
%Y A326879 Connected connectedness systems are counted by A326868, with unlabeled version A326869.
%Y A326879 Connected connectedness systems without singletons are counted by A072447.
%Y A326879 The not necessarily connected case is A326872.
%Y A326879 Cf. A029931, A048793, A072445, A072446, A326031, A326749, A326753, A326866, A326867, A326870, A326876.
%K A326879 nonn
%O A326879 1,3
%A A326879 _Gus Wiseman_, Jul 29 2019