|
|
A329560
|
|
BII-numbers of antichains of sets with empty intersection.
|
|
4
|
|
|
0, 3, 9, 10, 11, 12, 18, 33, 52, 129, 130, 131, 132, 136, 137, 138, 139, 140, 144, 146, 148, 160, 161, 164, 176, 180, 192, 258, 264, 266, 268, 274, 288, 292, 304, 308, 513, 520, 521, 524, 528, 532, 545, 560, 564, 772, 776, 780, 784, 788, 800, 804, 816, 820, 832
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
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 set-system (finite set of finite nonempty sets of positive integers) 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.
A set-system is an antichain if no edge is a proper subset of any other.
Empty intersection means there is no vertex in common to all the edges
|
|
LINKS
|
|
|
EXAMPLE
|
The sequence of terms together with their binary expansions and corresponding set-systems begins:
0: 0 ~ {}
3: 11 ~ {{1},{2}}
9: 1001 ~ {{1},{3}}
10: 1010 ~ {{2},{3}}
11: 1011 ~ {{1},{2},{3}}
12: 1100 ~ {{1,2},{3}}
18: 10010 ~ {{2},{1,3}}
33: 100001 ~ {{1},{2,3}}
52: 110100 ~ {{1,2},{1,3},{2,3}}
129: 10000001 ~ {{1},{4}}
130: 10000010 ~ {{2},{4}}
131: 10000011 ~ {{1},{2},{4}}
132: 10000100 ~ {{1,2},{4}}
136: 10001000 ~ {{3},{4}}
137: 10001001 ~ {{1},{3},{4}}
138: 10001010 ~ {{2},{3},{4}}
139: 10001011 ~ {{2},{3},{4}}
140: 10001100 ~ {{1,2},{3},{4}}
144: 10010000 ~ {{1,3},{4}}
146: 10010010 ~ {{2},{1,3},{4}}
148: 10010100 ~ {{1,2},{1,3},{4}}
|
|
MATHEMATICA
|
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 1}];
Select[Range[0, 100], #==0||Intersection@@bpe/@bpe[#]=={}&&stableQ[bpe/@bpe[#], SubsetQ]&]
|
|
CROSSREFS
|
BII-numbers of intersecting set-systems with empty intersecting are A326912.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|