OFFSET
1,2
COMMENTS
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.
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.
The enumeration of these set-systems by number of covered vertices is given by A326877.
LINKS
Gus Wiseman, Every Clutter Is a Tree of Blobs, The Mathematica Journal, Vol. 19, 2017.
EXAMPLE
The sequence of all connectedness systems without singletons together with their BII-numbers begins:
0: {}
4: {{1,2}}
16: {{1,3}}
32: {{2,3}}
64: {{1,2,3}}
68: {{1,2},{1,2,3}}
80: {{1,3},{1,2,3}}
84: {{1,2},{1,3},{1,2,3}}
96: {{2,3},{1,2,3}}
100: {{1,2},{2,3},{1,2,3}}
112: {{1,3},{2,3},{1,2,3}}
116: {{1,2},{1,3},{2,3},{1,2,3}}
256: {{1,4}}
288: {{2,3},{1,4}}
512: {{2,4}}
528: {{1,3},{2,4}}
1024: {{1,2,4}}
1028: {{1,2},{1,2,4}}
1280: {{1,4},{1,2,4}}
1284: {{1,2},{1,4},{1,2,4}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
connnosQ[eds_]:=!MemberQ[Length/@eds, 1]&&SubsetQ[eds, Union@@@Select[Tuples[eds, 2], Intersection@@#!={}&]];
Select[Range[0, 1000], connnosQ[bpe/@bpe[#]]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jul 29 2019
STATUS
approved